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You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
In parallel... | [
"7. Since \\\\dfrac{AU}{UD} = \\\\dfrac{AE}{DN} = \\\\dfrac{2a}{b} and AD = DM = b, we obtain that \\\\dfrac{AU}{UM} = \\\\dfrac{b}{a + b}.\n8. Analogously, \\\\dfrac{AV}{VB} = \\\\dfrac{b}{a + b}.\n9. By Ceva's Theorem, we obtain that \\\\dfrac{ET}{TM} = \\\\dfrac{b}{a} = \\\\dfrac{AE}{AM}.\n10. This concludes our... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given posit... | [
"5. Combine the terms over a common denominator:\n \\[\n \\sum_{cyc} \\frac{(x + 1)(3 + x)^2 - 32x}{32(3 + x)^2} \\geq 0\n \\]\n6. Expand and simplify the numerator:\n \\[\n (x + 1)(3 + x)^2 - 32x = (x + 1)(9 + 6x + x^2) - 32x = 9x + 6x^2 + x^3 + 9 + 6x + x^2 - 32x = x^3 + 7x^2 - 17x + 9\n \\]\n7. Facto... | A |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What are the ... | [
"### 2. **Powdery Coating on Upper Leaf Surface** \n - A **white, talc-like powder** forms primarily on the **upper leaf surface**, easily rubbing off. \n - Growth is superficial and does not cause corresponding patches on the opposite side. \n\n### 3. **Fruit Spotting and Surface Russeting** \n - **Small... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $x, y, ... | [
"9. We can rewrite the inequality as:\n \\[\n \\sum_{\\text{cyc}} (x^4 + 8x^2y^2) \\geq \\sum_{\\text{cyc}} (3x^3y + 3x^3z + 4xyz)\n \\]\n\n10. To prove this, we use the rearrangement inequality:\n \\[\n \\sum x^4 + 6\\sum x^2y^2 \\geq 4\\sum x^3y + 4xyz\n \\]\n\n11. Matching coefficients yields:\n ... | C |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"11. Let x' = -x. Then \\{x\\} = 1 - \\{x'\\} and -\\left\\lfloor x \\right\\rfloor = \\left\\lfloor x' \\right\\rfloor + 1.\n12. The inequality 8\\{x\\} > -x becomes 8(1 - \\{x'\\}) > x', which simplifies to 9\\{x'\\} + \\left\\lfloor x' \\right\\rfloor \\le 8.\n13. Since \\left\\lfloor x' \\right\\rfloor is even,... | F |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Derive the fo... | [
"V = \\int_{0}^{h} \\left( \\int_{0}^{2\\pi} \\left( \\int_{0}^{R} dr \\right) d\\theta \\right) dz = \\int_{0}^{h} \\left( \\int_{0}^{2\\pi} R \\, d\\theta \\right) dz = \\int_{0}^{h} 2\\pi R \\, dz = 2\\pi R h.",
"V = \\int_{0}^{h} \\left( \\int_{0}^{2\\pi} \\left( \\int_{0}^{R} r \\, dr \\right) d\\theta \\rig... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $\trian... | [
"2. Apply the Desargues' Involution Theorem (DDIT) on quadrilateral $ABCP$. 3. By DDIT, there exists an involution that swaps the pairs of lines $(\\overline{PA}, \\overline{PB})$, $(\\overline{PB'}, \\overline{PC'})$, and $(\\overline{PC}, \\overline{PA'})$. 4. Note that the pairs $(\\overline{PA}, \\overline{PB})... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $a, b, ... | [
"8. Rearranging terms, we get:\n \\[ 6 \\left( a^3b + a^3c + b^3c + b^3a + c^3a + c^3b \\right) + 3 \\left( a^2b^2 + b^2c^2 + c^2a^2 \\right) + 12abc - 4(a^3b + a^3c + b^3a + b^3c + c^3a + c^3b) - 6(a^2b^2 + a^2c^2 + b^2c^2) - 12abc \\leq a^4 + b^4 + c^4 \\]\n\n9. Simplifying further, we get:\n \\[ 4 \\left( a^... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let a man w... | [
"4. By the pigeonhole principle, after taking $\\left\\lceil \\frac{1}{\\epsilon} \\right\\rceil + 1$ steps, the man must have visited at least two points that are a distance of strictly less than $\\epsilon$ apart on the circle.\n5. Let these two points be $x_n$ and $x_m$, where $m$ and $n$ are the step numbers co... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that ... | [
"14. By the inductive hypothesis, x^k + x^{k-2} + \\ldots + \\frac{1}{x^{k-2}} + \\frac{1}{x^k} \\ge k + 1.\n15. Moreover, x^{k+2} + \\frac{1}{x^{k+2}} = 2.\n16. Adding, we find x^{k+2} + x^k + x^{k-2} + \\ldots + \\frac{1}{x^{k-2}} + \\frac{1}{x^k} + \\frac{1}{x^{k+2}} = (k + 1) + 2.\n17. Therefore, x^{k+2} + x^k ... | C |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Design an alg... | [
"2. **Dynamic Programming Update:** \n For each intermediate vertex k (from 1 to n), only update pairs with i < k < j: \n If i < k < j then A(i,j) = min { A(i,j), A(i,k) + A(k,j) } else leave A(i,j) unchanged.\n This ensures improved efficiency by avoiding unnecessary updates.\n\n3. **Handling Negative Weig... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
For positiv... | [
"9. By Cauchy–Schwarz,\n \\[ \\left( \\sum \\frac{1}{x} \\right)^2 \\geq \\sum \\frac{1}{xy} \\]\n\n10. From $x + y + z = xyz$, we have:\n \\[ \\sum \\frac{1}{xy} = 1 \\]\n\n11. Thus,\n \\[ \\left( \\sum \\frac{1}{x} \\right)^2 \\geq 1 \\]\n\n12. Therefore,\n \\[ \\sum \\frac{1}{x} \\geq 1. \\]",
"9. B... | C |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
A man follows... | [
"1. Shower Schedule:\n- If he starts on Day 1, he showers 5 times in Week 1 and 4 times in Week 2.\n- For example:\n - Week 1: Shower on Day 1, 2, 4, 6, 7 (5 showers).\n - Week 2: Shower on Day 1, 3, 5, 7 (4 showers).",
"1. Shower Schedule:\n- If he showers every other day, he showers 3-4 times a week (dependin... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $a_n = ... | [
"12. Let $g(x) = \\left(1 + \\frac{1}{x}\\right)^{x+2}$. The inequality transforms into $g(n+1) - g(n) < -\\frac{1}{n(n-1)}$.\n13. Since $g$ is decreasing and concave, it suffices to show $g'(n) < -\\frac{1}{n(n-1)}$.\n14. We compute $g'(n) = g(n)\\!\\left(\\ln\\left(1 + \\frac{1}{n}\\right) - \\frac{n+2}{n(n+1)}\\... | G |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
In a finite... | [
"3. At step i + 1, pick v_{i + 1} so that it is new globally (never seen before).\n4. Continue until you cannot proceed; by finiteness this occurs before any vertex repeats.\n5. Each step offers at least k neighbors, so the path length can be extended arbitrarily without repetition.\n6. Since the process stops only... | G |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Prove that ... | [
"8. Step 8: Use the identity \\(\\sin A \\sin B = \\frac{1}{2} [\\cos (A-B) - \\cos (A+B)]\\):\\n\\[1 + \\cos C \\ge \\cos (A-B) - \\cos (A+B)\\]\\n\\n9. Step 9: Since \\(\\cos (A+B) = -\\cos C\\):\\n\\[1 + \\cos C \\ge \\cos (A-B) + \\cos C\\]\\n\\n10. Step 10: Subtract \\(\\cos C\\) from both sides:\\n\\[1 \\ge \... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $x, y, ... | [
"5. Substitute the given substitutions:\n \\sum_{\\text{cyc}} \\left(2 + \\frac{3-a}{a} - 2\\sqrt{\\frac{(3-a)(3-b)}{ab}}\\right)\n6. Simplify the expression inside the summation:\n \\sum_{\\text{cyc}} \\left(2 + \\frac{b+c}{a} - 2\\sqrt{\\frac{(b+c)(a+c)}{ab}}\\right)\n7. Further simplify:\n \\sum_{\\text{cy... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Solve the i... | [
"2. Rewrite the inequality x\\sqrt{12-x} + (11-x)\\sqrt{x+1} \\geq 25 in terms of t:\n \\[x\\sqrt{12-x} + (11-x)\\sqrt{x+1} - 25 = \\frac{1}{2}(t-5)^2(t+2)\\]\n3. For the inequality to hold, the expression \\frac{1}{2}(t-5)^2(t+2) must be non-negative.\n4. Since (t-5)^2 \\geq 0, the sign is determined by t+2.\n5.... | C |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given a tri... | [
"2. Note that triangles AEF and ABC are similar, i.e., triangle AEF ~ triangle ABC.\n3. Similarly, triangles HEF and DBC are also similar, i.e., triangle HEF ~ triangle DBC.\n4. From the similarity of triangle AEF with triangle ABC, and triangle HEF with triangle DBC, it follows that AH and AD are isogonal with res... | A |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
How does coro... | [
"Coronary blood flow is greater during diastole than during systole. This is because the contracting ventricular myocardium compresses the coronary vessels during systole, reducing blood flow. During diastole, when the heart relaxes, this compression is relieved, allowing for increased blood flow to the myocardium.... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given a par... | [
"6. From the angle equalities, we get ∠BEU = ∠BFV, so E and F lie on the circle with diameter UV.\n7. Because (J) is tangent to CB and CD at U and V, it has UV as diameter.\n8. Hence (J) passes through E and F.\n9. Therefore E and F have zero power with respect to (J).\n10. This implies the desired circle is (J).",... | E |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given that ad... | [
"2. Adenosine antagonists primarily increase GABAergic inhibition of dopaminergic neurons.\n3. As a result, blocking adenosine receptors would lower dopamine output and aggravate motor symptoms.",
"2. Adenosine receptors in the central nervous system indirectly inhibit dopamine release.\n3. Blocking adenosine rec... | B |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $\{a_n\... | [
"Substituting these into the sum, we get \\sum_{k=n+1}^{2n+1} a_k = \\frac{n+1}{2}((a_1 + nd) + (a_1 + 2nd)) = \\frac{n+1}{2}(2a_{n+1} + a_1).\nApplying the Cauchy–Schwarz inequality, \\frac{n+1}{2}(2a_{n+1} + a_1) \\le \\frac{n+1}{2}\\sqrt{(2^2 + 1^2)(a_{n+1}^2 + a_1^2)}.\nSince a_1^2 + a_{n+1}^2 \\le \\frac{2}{5}... | D |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine t... | [
"8. Set the discriminant to zero: (8m)^2 - 4(4m^2 + 1)(3) = 0.\n9. Simplify: 64m^2 - 12(4m^2 + 1) = 0.\n10. Expand: 64m^2 - 48m^2 - 12m^2 = 0.\n11. Combine: 4m^2 = 0.\n12. Solve: m^2 = 0.\n13. Conclude: m^2 = 0.\n14. The value of m^2 for which the line intersects the ellipse at exactly one point is 0.",
"8. Set t... | C |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $a, b, ... | [
"Since:\n \\[\n a + \\frac{a^2}{b + c - a} = \\frac{a(a + b + c)}{b + c - a},\n \\]\n the inequality can be rewritten as:\n \\[\n \\frac{2abc(a + b + c)}{(b + c - a)(c + a - b)(a + b - c)} \\ge (a + b + c) + (a + b + c) \\left( \\frac{a}{b + c - a} + \\frac{b}{c + a - b} + \\frac{c}{a + b - c} \\right).... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Let $a$ and... | [
"6. Apply Cauchy-Schwarz with three terms using weights 1/2:\n (a^2 + 1/2 + 1/2)(b^2 + 1/2 + 1/2) >= ( a/sqrt(2) + b/sqrt(2) + 1 )^2\n7. Simplify the right-hand side:\n ( (a + b)/sqrt(2) + 1 )^2\n8. Substitute this into the expression:\n ((a^2 + 1)(b^2 + 1))/(a + b) >= ( ( (a + b)/sqrt(2) + 1 )^2 )/(a + b)\n9... | B |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Describe a me... | [
"4. Affine Normalization:\n- Perform an affine transformation using the square root of C:\nA = C^{1/2}, X'' = A X'.\n- This produces a transformed shape whose covariance remains C rather than the identity.\n\n5. Fitting the Shape into a Unit Circle:\n- Solve the minimization problem:\nmin_{alpha} (1/n) sum_i || a... | F |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given $x, y... | [
"5. We have:\n \n LHS = 1 - sqrt(8xyz/((x+y)(y+z)(z+x))) = ((x+y)(y+z)(z+x) - 8xyz)/(1 + sqrt(8xyz/((x+y)(y+z)(z+x)))).\n\n6. Using the lemma, we get:\n \n LHS <= ((x-y)(y-z)(x-z))/(1 + sqrt(8xyz/((x+y)(y+z)(z+x)))).\n\n7. For the right-hand side (RHS), we have:\n \n RHS = prod_cyc ( 1 - sqrt(2x/(x+y)) ... | G |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What is clini... | [
"1. If there is visible recession, CAL = (Recession depth) + (Pocket/sulcus depth).\n2. If there is no visible recession, CAL = (Distance from CEJ to the base of the pocket).",
"1. If there is visible recession, CAL = (Pocket/sulcus depth) + (Distance from gingival margin to CEJ).\n2. If there is no visible reces... | A |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Based on the ... | [
"2. **High-sodium fast food at every meal**: Depending on takeout multiple times daily exposes children to excessive salt and fat, which was not part of Lindsey's reported routine.\n3. **Allowing caffeinated soda regularly**: Caffeine and high sugar in soda are inappropriate for toddlers, but this specific issue wa... | G |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
How does tiag... | [
"However, tiagabine increases the plasma concentration of SSRIs and tricyclics through CYP inhibition. This pharmacokinetic interaction can lead to increased serotonergic side effects.",
"However, tiagabine augments GABAergic tone enough to counteract any proconvulsant effects of SSRIs or tricyclics. Net clinical... | E |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Determine whe... | [
"The graphs are not isomorphic because one has a cut-vertex while the other does not. Structural differences of this kind rule out any isomorphism.",
"The graphs are not isomorphic because G has a vertex of degree 3 while G' has none. This difference in maximum degree prevents any isomorphism.",
"The graphs are... | C |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given that ... | [
"3. Apply the average lower bound:\n max(a + b + c, c + d + e, e + f + g) >= ((a + b + c) + (c + d + e) + (e + f + g))/3\n4. Grouping terms, note c and e are double-counted; subtract them once:\n ((...))/3 = (a + b + c + d + e + f + g - c - e)/3\n5. Substitute a + b + c + d + e + f + g = 1:\n (1 - c - e)/3\n6... | E |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What are the ... | [
"3. ER and PR negativity: Absence of hormone receptors is associated with a better prognosis.\n4. Her-2-Neu negativity: Lack of Her-2-Neu expression predicts a worse prognosis.\n5. Lymphovascular invasion: Absence indicates a poorer prognosis.\n6. Proliferative rate: Low proliferative index is associated with worse... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
In what perce... | [
"Step 4: Conclusion\nBased on aggregated data:\n- Approximately 20-25% of patients (average across populations) exhibit an inter-arm SBP difference ≥10 mmHg when measured sequentially.\n- In high-risk populations (e.g., cardiovascular disease), this may rise to 30% or higher.\nThus, the most widely cited estimate i... | B |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Use completin... | [
"Half of -2 is -1, and (-1)^2 = 1.\nf(x) = -2(x^2 - 2x + 1 - 1) + 7\nf(x) = -2((x^2 - 2x + 1) - 1) + 7\nf(x) = -2(x - 1)^2 + 2 + 7 (Distribute the -2)\nf(x) = -2(x - 1)^2 + 9",
"Half of -2 is -1, and (-1)^2 = 1.\nf(x) = -2(x^2 - 2x + 1 - 1) + 7\nf(x) = -2((x - 1)^2 - 1) + 7\nf(x) = (x - 1)^2 + 2 + 7\nf(x) = (x - ... | A |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given the e... | [
"3. Since Y is normal, hat{beta} is normal with covariance sigma^2 (X^T X).\n4. Hence, hat{Y} = X hat{beta} is normal.\n5. The distribution is hat{Y} ~ N(X beta, sigma^2 X (X^T X) X^T).",
"3. Since Y is normal, hat{beta} is normal.\n4. Therefore, hat{Y} = X hat{beta} is normal with variance estimated by s^2.\n5. ... | F |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What are the ... | [
"Girls who receive only cranial irradiation without abdominal exposure are at minimal risk for gonadal failure and typically experience normal timing of puberty. In these patients, routine monitoring is limited to prolactin and thyroid-stimulating hormone, with menstrual history generally unnecessary.",
"Radiatio... | H |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Show that for... | [
"2. **Evaluate [\\mathbf{r} \\cdot \\mathbf{p}, \\frac{\\mathbf{p}^2}{2m}]:**\n - With \\([\\mathbf{r}_i, \\mathbf{p}_j] = i\\hbar \\delta_{ij}\\), we have \\([\\mathbf{r}_i, \\mathbf{p}_j^2] = 2 i\\hbar \\delta_{ij} \\mathbf{p}_j\\), leading to \\([\\mathbf{r}_i \\mathbf{p}_i, \\mathbf{p}_j^2] = 2 i\\hbar \\delt... | E |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Define a cyli... | [
"1. **Cross-Section (\\(\\Omega\\))**: A 2D shape in a plane.\n2. **Curvilinear Sweep**: \\(\\Omega\\) is moved along a curved axis (e.g., an arc or helix) rather than a straight line; the total arc length is taken as the height.\n3. **Resulting Solid**: The union of the swept shapes yields a curved-axis tube that ... | E |
You are given a math problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the mathematical context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
Given $a, b... | [
"12. Rewriting the left-hand side, we have:\n \\[\n \\left(\\log_{\\frac{1}{m}} \\left(\\frac{m + n}{2}\\right)\\right) \\left(\\log_{\\frac{1}{n}} \\left(\\frac{m + n}{2}\\right)\\right) \\geq 1\n \\]\n13. Since \\(\\log_b x = -\\log_b(1/x)\\), we get a sum after expanding:\n \\[\n -\\log_{\\frac{1}... | D |
You are given a STEM problem and its solution, with some steps replaced by [MASK]. Several candidate options are provided for the missing steps. Carefully analyze the scientific context and select the option that best fills in the [MASK]. Output the letter of the correct option in \boxed{}.
**Question:**
What is the p... | [
"2. **Fever Induction**: Resetting of the hypothalamic set point by pyrogens to elevate body temperature during infection.\n3. **Acute-Phase Response**: Hepatic synthesis of acute-phase proteins (e.g., CRP, fibrinogen) under IL-6 signaling.",
"2. **Antibody Production**: Differentiation of B cells into plasma cel... | G |
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