{"year": "2009", "tier": "T4", "problem_label": "1", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$.", "solution": "47", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n1. [3]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "1", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$.", "solution": "We can factor the equation as $\\left(a-b^{2}\\right)\\left(a+b^{2}\\right)=41 \\cdot 49$, from which it is evident that $a=45$ and $b=2$ is a possible solution. By examining the factors of 2009 , one can see that there are no other solutions.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n1. [3]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "2", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\\log _{2}(S)$ ?", "solution": "1004", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n2. [3]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "2", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\\log _{2}(S)$ ?", "solution": "The sum of all the coefficients is $(1+i)^{2009}$, and the sum of the real coefficients is the real part of this, which is $\\frac{1}{2}\\left((1+i)^{2009}+(1-i)^{2009}\\right)=2^{1004}$. Thus $\\log _{2}(S)=1004$.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n2. [3]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "3", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $\\tan x+\\tan y=4$ and $\\cot x+\\cot y=5$, compute $\\tan (x+y)$.", "solution": "20", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n3. [4]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "3", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $\\tan x+\\tan y=4$ and $\\cot x+\\cot y=5$, compute $\\tan (x+y)$.", "solution": "We have $\\cot x+\\cot y=\\frac{\\tan x+\\tan y}{\\tan x \\tan y}$, so $\\tan x \\tan y=\\frac{4}{5}$. Thus, by the tan sum formula, $\\tan (x+y)=\\frac{\\tan x+\\tan y}{1-\\tan x \\tan y}=20$.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n3. [4]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "4", "problem_type": "Algebra", "exam": "HMMT", "problem": "Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$.", "solution": "$\\quad-6$", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n4. [4]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "4", "problem_type": "Algebra", "exam": "HMMT", "problem": "Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$.", "solution": "Since $x+1$ divides $x^{2}+a x+b$ and the constant term is $b$, we have $x^{2}+a x+b=(x+1)(x+b)$, and similarly $x^{2}+b x+c=(x+1)(x+c)$. Therefore, $a=b+1=c+2$. Furthermore, the least common multiple of the two polynomials is $(x+1)(x+b)(x+b-1)=x^{3}-4 x^{2}+x+6$, so $b=-2$. Thus $a=-1$ and $c=-3$, and $a+b+c=-6$.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n4. [4]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "5", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}$.", "solution": "$\\quad-2$", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n5. [4]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "5", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}$.", "solution": "We can substitute $x=y-1$ to obtain a polynomial having roots $a+1, b+1, c+1$, namely, $(y-1)^{3}-(y-1)+1=y^{3}-3 y^{2}+2 y+1$. The sum of the reciprocals of the roots of this polynomial is, by Viete's formulas, $\\frac{2}{-1}=-2$.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n5. [4]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "6", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $x$ and $y$ be positive real numbers and $\\theta$ an angle such that $\\theta \\neq \\frac{\\pi}{2} n$ for any integer $n$. Suppose\n\n$$\n\\frac{\\sin \\theta}{x}=\\frac{\\cos \\theta}{y}\n$$\n\nand\n\n$$\n\\frac{\\cos ^{4} \\theta}{x^{4}}+\\frac{\\sin ^{4} \\theta}{y^{4}}=\\frac{97 \\sin 2 \\theta}{x^{3} y+y^{3} x}\n$$\n\nCompute $\\frac{x}{y}+\\frac{y}{x}$.", "solution": "4", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n6. [5]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "6", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $x$ and $y$ be positive real numbers and $\\theta$ an angle such that $\\theta \\neq \\frac{\\pi}{2} n$ for any integer $n$. Suppose\n\n$$\n\\frac{\\sin \\theta}{x}=\\frac{\\cos \\theta}{y}\n$$\n\nand\n\n$$\n\\frac{\\cos ^{4} \\theta}{x^{4}}+\\frac{\\sin ^{4} \\theta}{y^{4}}=\\frac{97 \\sin 2 \\theta}{x^{3} y+y^{3} x}\n$$\n\nCompute $\\frac{x}{y}+\\frac{y}{x}$.", "solution": "From the first relation, there exists a real number $k$ such that $x=k \\sin \\theta$ and $y=k \\cos \\theta$. Then we have\n\n$$\n\\frac{\\cos ^{4} \\theta}{\\sin ^{4} \\theta}+\\frac{\\sin ^{4} \\theta}{\\cos ^{4} \\theta}=\\frac{194 \\sin \\theta \\cos \\theta}{\\sin \\theta \\cos \\theta\\left(\\cos ^{2} \\theta+\\sin ^{2} \\theta\\right)}=194\n$$\n\nNotice that if $t=\\frac{x}{y}+\\frac{y}{x}$ then $\\left(t^{2}-2\\right)^{2}-2=\\frac{\\cos ^{4} \\theta}{\\sin ^{4} \\theta}+\\frac{\\sin ^{4} \\theta}{\\cos ^{4} \\theta}=194$ and so $t=4$.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n6. [5]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "7", "problem_type": "Algebra", "exam": "HMMT", "problem": "Simplify the product\n\n$$\n\\prod_{m=1}^{100} \\prod_{n=1}^{100} \\frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}\n$$\n\nExpress your answer in terms of $x$.", "solution": "$x^{9900}\\left(\\frac{1+x^{100}}{2}\\right)^{2}\\left(\\right.$ OR $\\left.\\frac{1}{4} x^{9900}+\\frac{1}{2} x^{10000}+\\frac{1}{4} x^{10100}\\right)$", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n7. [5]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "7", "problem_type": "Algebra", "exam": "HMMT", "problem": "Simplify the product\n\n$$\n\\prod_{m=1}^{100} \\prod_{n=1}^{100} \\frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}\n$$\n\nExpress your answer in terms of $x$.", "solution": "We notice that the numerator and denominator of each term factors, so the product is equal to\n\n$$\n\\prod_{m=1}^{100} \\prod_{n=1}^{100} \\frac{\\left(x^{m}+x^{n+1}\\right)\\left(x^{m+1}+x^{n}\\right)}{\\left(x^{m}+x^{n}\\right)^{2}}\n$$\n\nEach term of the numerator cancels with a term of the denominator except for those of the form $\\left(x^{m}+x^{101}\\right)$ and $\\left(x^{101}+x^{n}\\right)$ for $m, n=1, \\ldots, 100$, and the terms in the denominator which remain are of the form $\\left(x^{1}+x^{n}\\right)$ and $\\left(x^{1}+x^{m}\\right)$ for $m, n=1, \\ldots, 100$. Thus the product simplifies to\n\n$$\n\\left(\\prod_{m=1}^{100} \\frac{x^{m}+x^{101}}{x^{1}+x^{m}}\\right)^{2}\n$$\n\nReversing the order of the factors of the numerator, we find this is equal to\n\n$$\n\\begin{aligned}\n\\left(\\prod_{m=1}^{100} \\frac{x^{101-m}+x^{101}}{x^{1}+x^{m}}\\right)^{2} & =\\left(\\prod_{m=1}^{100} x^{100-m} \\frac{x^{1}+x^{m+1}}{x^{1}+x^{m}}\\right)^{2} \\\\\n& =\\left(\\frac{x^{1}+x^{1} 01}{x^{1}+x^{1}} \\prod_{m=1}^{100} x^{100-m}\\right)^{2} \\\\\n& =\\left(x^{\\frac{99 \\cdot 100}{2}}\\right)^{2}\\left(\\frac{1+x^{100}}{2}\\right)^{2}\n\\end{aligned}\n$$\n\nas desired.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n7. [5]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "8", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$.", "solution": "20.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n8. [7]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "8", "problem_type": "Algebra", "exam": "HMMT", "problem": "If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$.", "solution": "We have $a x^{3}+b y^{3}=16$, so $\\left(a x^{3}+b y^{3}\\right)(x+y)=16(x+y)$ and thus\n\n$$\na x^{4}+b y^{4}+x y\\left(a x^{2}+b y^{2}\\right)=16(x+y)\n$$\n\nIt follows that\n\n$$\n42+7 x y=16(x+y)\n$$\n\nFrom $a x^{2}+b y^{2}=7$, we have $\\left(a x^{2}+b y^{2}\\right)(x+y)=7(x+y)$ so $a x^{3}+b y^{3}+x y\\left(a x^{2}+b y^{2}\\right)=7(x+y)$. This simplifies to\n\n$$\n16+3 x y=7(x+y)\n$$\n\nWe can now solve for $x+y$ and $x y$ from (1) and (2) to find $x+y=-14$ and $x y=-38$. Thus we have $\\left(a x^{4}+b y^{4}\\right)(x+y)=42(x+y)$, and so $a x^{5}+b y^{5}+x y\\left(a x^{3}+b y^{3}\\right)=42(x+y)$. Finally, it follows that $a x^{5}+b y^{5}=42(x+y)-16 x y=20$ as desired.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n8. [7]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "9", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $\\left|z_{a} z_{b}+z_{c} z_{d}\\right|$ where $\\{a, b, c, d\\}=\\{1,2,3,4\\}$.", "solution": "8", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n9. [7]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "9", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $\\left|z_{a} z_{b}+z_{c} z_{d}\\right|$ where $\\{a, b, c, d\\}=\\{1,2,3,4\\}$.", "solution": "Note that $\\frac{1}{16} f(2 x)=x^{4}+7 x^{3}+13 x^{2}+7 x+1$. Because the coefficients of this polynomial are symmetric, if $r$ is a root of $f(x)$ then $\\frac{4}{r}$ is as well. Further, $f(-1)=-1$ and $f(-2)=16$ so $f(x)$ has two distinct roots on $(-2,0)$ and two more roots on $(-\\infty,-2)$. Now, if $\\sigma$ is a permutation of $\\{1,2,3,4\\}$ :\n$\\left|z_{\\sigma(1)} z_{\\sigma(2)}+z_{\\sigma(3)} z_{\\sigma(4)}\\right| \\leq \\frac{1}{2}\\left(z_{\\sigma(1)} z_{\\sigma(2)}+z_{\\sigma(3)} z_{\\sigma(4)}+z_{\\sigma(4)} z_{\\sigma(3)}+z_{\\sigma(2)} z_{\\sigma(1)}\\right)$\nLet the roots be ordered $z_{1} \\leq z_{2} \\leq z_{3} \\leq z_{4}$, then by rearrangement the last expression is at least:\n$\\frac{1}{2}\\left(z_{1} z_{4}+z_{2} z_{3}+z_{3} z_{2}+z_{4} z_{1}\\right)$\nSince the roots come in pairs $z_{1} z_{4}=z_{2} z_{3}=4$, our expression is minimized when $\\sigma(1)=1, \\sigma(2)=$ $4, \\sigma(3)=3, \\sigma(4)=2$ and its minimum value is 8 .", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n9. [7]", "solution_match": "\nSolution: "}}
{"year": "2009", "tier": "T4", "problem_label": "10", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle?", "solution": "$\\left[\\frac{\\sqrt{3}}{3}, 1\\right]$", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n10. [8]", "solution_match": "\nAnswer: "}}
{"year": "2009", "tier": "T4", "problem_label": "10", "problem_type": "Algebra", "exam": "HMMT", "problem": "Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle?", "solution": "Say we have four points $(a, f(a)),(b, f(b)),(c, f(c)),(d, f(d))$ on the curve which form a rectangle. If we interpolate a cubic through these points, that cubic will be symmetric around the center of the rectangle. But the unique cubic through the four points is $f(x)$, and $f(x)$ has only one point of symmetry, the point $(0,0)$.\nSo every rectangle with all four points on $f(x)$ is of the form $(a, f(a)),(b, f(b)),(-a, f(-a)),(-b, f(-b))$, and without loss of generality we let $a, b>0$. Then for any choice of $a$ and $b$ these points form a parallelogram, which is a rectangle if and only if the distance from $(a, f(a))$ to $(0,0)$ is equal to the distance from $(b, f(b))$ to $(0,0)$. Let $g(x)=x^{2}+(f(x))^{2}=4 x^{6}-8 x^{4}+5 x^{2}$, and consider $g(x)$ restricted to $x \\geq 0$. We are looking for all the values of $a$ such that $g(x)=g(a)$ has solutions other than $a$.\nNote that $g(x)=h\\left(x^{2}\\right)$ where $h(x)=4 x^{3}-8 x^{2}+5 x$. This polynomial $h(x)$ has a relative maximum of 1 at $x=\\frac{1}{2}$ and a relative minimum of $25 / 27$ at $x=\\frac{5}{6}$. Thus the polynomial $h(x)-h(1 / 2)$ has the double root $1 / 2$ and factors as $\\left(4 x^{2}-4 x+1\\right)(x-1)$, the largest possible value of $a^{2}$ for which $h\\left(x^{2}\\right)=h\\left(a^{2}\\right)$ is $a^{2}=1$, or $a=1$. The smallest such value is that which evaluates to $25 / 27$ other than $5 / 6$, which is similarly found to be $a^{2}=1 / 3$, or $a=\\frac{\\sqrt{3}}{3}$. Thus, for $a$ in the range $\\frac{\\sqrt{3}}{3} \\leq a \\leq 1$ the equation $g(x)=g(a)$ has nontrivial solutions and hence an inscribed rectangle exists.", "metadata": {"resource_path": "HarvardMIT/segmented/en-122-2009-feb-alg-solutions.jsonl", "problem_match": "\n10. [8]", "solution_match": "\nSolution: "}}