prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square with side length 1 is rotated about one vertex by an angle of $\\alpha,$ where $0^\\circ < \\alpha < 90^\\circ$ and $... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\\left(\\frac{1}{a+b}+\\frac{1}... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of $\\triangle PQR$ and $\\triangle STU$ has an area of $1.$ In $\\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 2.$ Find the maximum value of\n\\[(x^2 - xy + y^2)(x... | 499 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no t... | 107 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the s... | 53 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0),$ $B = (0,4,0)$, $C = (0,0,2),$ and $D = (0,0,0).$ F... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,$ $b,$ $c,$ $d$ be positive real numbers. Find the minimum value of\n\\[(1 + a)(1 + b)(1 + c)(1 + d) \\left( \\frac{1}... | 1051 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and ... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, triangle $ABC$ has been reflected over its median $\\overline{AM}$ to produce triangle $AB'C'$. If $AE ... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\sum_{j = 0}^\\infty \\sum_{k = 0}^\\infty 2^{-3k - j - (k + j)^2}.\\]The answer is in the form \frac{m}{n}, wher... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point with coordinates $(x,\\ y)$ is randomly selected such that $0\\leq x \\leq10$ and $0\\leq y \\leq10$. What is the prob... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ and $y$ be nonnegative real numbers such that $x + y = 1.$ Find the maximum value of\n\\[x^4 y + xy^4.\\]The answer i... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe height of a cylindrical pole is 12 feet and its circumference is 2 feet. A rope is attached to a point on the circumferenc... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKeiko tosses one penny and Ephraim tosses two pennies. What is the probability that Ephraim gets the same number of heads that... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\]The answer is in the form \... | 237 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-6x+5|$ and $y=\\frac{29}{4}-x$.The ans... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJonathon is given the following problem: Let $a$ and $b$ be positive real numbers such that $a + b = 1.$ Find the minimum val... | 27 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1, a_2, \\dots$ be a sequence defined by $a_1 = a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 1$. Find \\[\n\\sum_{n=1}^... | 12 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen ... | 177 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6 on each die are in the ratio $1:2:... | 71 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $AHI$ is equilateral. We know $\\overline{BC}$, $\\overline{DE}$ and $\\overline{FG}$ are all parallel to $\\overline... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, we have $AB = 24$ and $\\angle ADB =90^\\circ$. If $\\sin A = \\frac23$ and $\\sin C = \\frac13$, then ... | 34 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, angle $C$ is a right angle and $CB > CA$. Point $D$ is located on $\\overline{BC}$ so that angle $CAD$ is t... | 14 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be three unit vectors, such that the angle between any of them is the acut... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum of the function\n\\[\\frac{xy}{x^2 + y^2}\\]in the domain $\\frac{2}{5} \\le x \\le \\frac{1}{2}$ and $\\frac... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, othe... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the hept... | 18 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = BC$, and $\\overline{BD}$ is an altitude. Point $E$ is on the extension of $\\overline{AC}$ such that... | 53 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has side length 2. A semicircle with diameter $\\overline{AB}$ is constructed inside the square, and the tangent... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the pro... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = AC = 5$ and $BC = 6$. Let $O$ be the circumcenter of triangle $ABC$. Find the area of triangle $OBC... | 29 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\[\\begin{aligned} a &= \\sqrt{2}+\\sqrt{3}+\\sqrt{6}, \\\\ b &= -\\sqrt{2}+\\sqrt{3}+\\sqrt{6}, \\\\ c&= \\sqrt{2}-\\sqr... | 625 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that the angles of triangle $ABC$ satisfy\n\\[\\cos 3A + \\cos 3B + \\cos 3C = 1.\\]Two sides of the triangle have len... | 400 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$. What is the largest of the th... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $\\left(\\sqrt[4]{11}\\right)^{3x-3}=\\frac{1}{5}$, what is the value of $\\left(\\sqrt[4]{11}\\right)^{6x+2}$? Express you... | 146 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the fraction that equals $0.72\\overline{6}$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide t... | 259 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe orthocenter of triangle $ABC$ divides altitude $\\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calcul... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that\n\\[(a + bi)^n = (a ... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Lakers are playing the Celtics in the NBA finals. To win the NBA finals, a team must win 4 games before the other team doe... | 2347 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoe will randomly select two letters from the word CAMP, four letters from the word HERBS, and three letters from the word GLO... | 31 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,$ $b$ be two acute angles where $\\tan a = 5 \\tan b.$ Find the maximum possible value of $\\sin (a - b).$The answer i... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p(x)$ be the polynomial of degree at most 8 such that\n\\[p(n) = \\frac{1}{n}\\]for $n = 1,$ 2, 3, $\\dots,$ 9. Find $p(... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle ... | 3004 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point $(x,y)$ is randomly selected such that $0 \\le x \\le 3$ and $0 \\le y \\le 6$. What is the probability that $x+y \\le... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJar A has exactly four red buttons and eight blue buttons. Carla then removes the same number of red buttons as blue buttons ... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting poin... | 121 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = 9$, $BC = 12$, $AC = 15$, and $CD$ is the angle bisector. Find the length of $CD$.The answer is in t... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute \\[\\dfrac{2^3-1}{2^3+1}\\cdot\\dfrac{3^3-1}{3^3+1}\\cdot\\dfrac{4^3-1}{4^3+1}\\cdot\\dfrac{5^3-1}{5^3+1}\\cdot\\dfrac... | 106 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation\n\\[\n z^4 - 6z^2 + 8z + 1 = -4(z^3... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\sin^2 4^\\circ + \\sin^2 8^\\circ + \\sin^2 12^\\circ + \\dots + \\sin^2 176^\\circ.\\]The answer is in the form... | 47 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute $\\tan\\left(\\frac{\\pi}{7}\\right)\\tan\\left(\\frac{2\\pi}{7}\\right)\\tan\\left(\\frac{3\\pi}{7}\\right)$.The answ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ $z$ be real numbers such that\n\\begin{align*}\nx + y + z &= 4, \\\\\nx^2 + y^2 + z^2 &= 6.\n\\end{align*}Let $m... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSet $R$ is a set of rectangles such that (1) only the grid points shown here are used as vertices, (2) all sides are vertical ... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't ... | 91 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $A(0,0), B(9,6)$ and $C(6,12)$ are vertices of triangle $ABC$. Point $D$ is on segment $AB$ such that $2(AD) = DB$, poi... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\cos^2 0^\\circ + \\cos^2 1^\\circ + \\cos^2 2^\\circ + \\dots + \\cos^2 90^\\circ.\\]The answer is in the form \... | 93 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathbf{p}$ and $\\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $30^\\circ.$ Fi... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? Express y... | 241 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(r) = \\sum_{j=2}^{2008} \\frac{1}{j^r} = \\frac{1}{2^r}+ \\frac{1}{3^r}+ \\dots + \\frac{1}{2008^r}$. Find $\\sum_{k=2}... | 4015 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ and $y$ be positive real numbers such that $x + \\frac{1}{y} = 5.$ Find the minimum value of $\\frac{1}{x} + y.$The a... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the p... | 37 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other,... | 33 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers $A$ and $B$ have radii 3 and 8, respectively. A common internal tangent touches the circles at $C$ and $... | 47 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $x^3 - 4x^2 + 5x - \\frac{19}{10} = 0$ has real roots $r,$ $s,$ and $t.$ Find the length of the long diagonal of... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\tau = \\frac{1 + \\sqrt{5}}{2}.$ Find\n\\[\\sum_{n = 0}^\\infty \\frac{\\lfloor \\tau^n \\rceil}{2^n}.\\]Note: For a r... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe parametric curve $(x,y) = (\\cos 2t, \\cos^2 t),$ for real numbers $t,$ describes a line segment. Find the length of this... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of numbers $a_1,$ $a_2,$ $a_3,$ $a_4,$ $\\dots,$ has the property that each term after the first is one less than t... | 3269 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum value of the function\n\\[f(x) = \\sqrt{-x^2 + 4x + 21} - \\sqrt{-x^2 + 3x + 10}.\\]The answer is in the form... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the maximum value of\n\\[\\begin{vmatrix} 1 & 1 & 1 \\\\ 1 & 1 + \\sin \\theta & 1 \\\\ 1 + \\cos \\theta & 1 & 1 \\end{v... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn the complex plane, the parallelogram formed by the points 0, $z,$ $\\frac{1}{z},$ and $z + \\frac{1}{z}$ has area $\\frac{3... | 87 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram, $R$ is on $QS$ and $QR=8$. Also, $PR=12$, $\\angle PRQ=120^\\circ$, and $\\angle RPS = 90^\\circ$. What is the... | 99 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe ratio of losses to wins for Kyle's team is 3 to 2. If the team had played the same number of games, but had won twice as m... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\sum_{n = 1}^\\infty \\frac{2n + 1}{n(n + 1)(n + 2)}.\\]The answer is in the form \frac{m}{n}, where gcd(m, n) = ... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number... | 55 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum possible value of\n\\[\\sqrt{58 - 42x} + \\sqrt{149 - 140 \\sqrt{1 - x^2}}\\]where $-1 \\le x \\le 1.$The ans... | 110 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBen rolls two fair six-sided dice. What is the expected value of the larger of the two numbers rolled? Express your answer as ... | 197 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $z$ be a complex number that satisfies\n\\[|z - 3i| + |z - 4| = 5.\\]Find the minimum value of $|z|.$The answer is in the ... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTo take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the p... | 30 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's?The answer is in the form \fra... | 1717 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle is drawn on the exterior of an equilateral triangle so the hypotenuse of the ri... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensiv... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that $f(1) = 1$ and\n\\[f(xy + f(x)) = xf(y) + f(x)\\]for all real n... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA bag contains 4 balls. Each ball is a single color, and 3 distinct colors are represented in the bag. What is the probability... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\\triangle A... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAllen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. Wha... | 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?The answer is in t... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can ... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smal... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBilly and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a mu... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $y - x,$ $y - 2x,$ and $y - kx$ are all factors of\n\\[x^3 - 3x^2 y + p_1 xy^2 + p_2 y^3,\\]then find $k$.The answer is in ... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fr... | 31 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number ch... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation \\[2000x^6+100x^5+10x^3+x-2=0\\]has two real roots. Compute the square of the difference between them.The answer ... | 561 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is... | 52 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these n... | 10057 |
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