prompt listlengths 1 1 | label stringlengths 1 210 |
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"role": "user",
"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 the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. For exampl... | 93 |
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{
"role": "user",
"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 $E$ and $F$ lie on diagonal $\\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE ... | 30 |
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A = \\frac{1}{6}((\\log_2(3))^3-(\\log_2(6))^3-(\\log_2(12))^3+(\\log_2(24))^3)$. Compute $2^A$.\n\nR... | 72 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) has side lengths \\( AB=20 \\), \\( BC=15 \\), and \\( CA=7 \\). Let the altitudes ... | 15 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box of strawberries, containing $12$ strawberries total, costs $2$. A box of blueberries, containing $48... | 120 |
[
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of integer pairs \\((x, y)\\) such that \\((|x| - 2)^2 + (|y| - 2)^2 < 5\\).\n\nRemem... | 48 |
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 ar... | 872 |
[
{
"role": "user",
"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 $p$, $q$, and $r$ are nonzero integers satisfying $p^2+q^2 = r^2$, compute the smallest possible value ... | 4 |
[
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"role": "user",
"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\nWe are given a $5 \\times 5$ square grid, divided into $1 \\times 1$ tiles. Two tiles are called **linked*... | 10 |
[
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"role": "user",
"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 land of Nyemo, the unit of currency is called a quack. The citizens use coins that are worth $1$, $... | 82 |
[
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and... | 1 |
[
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"role": "user",
"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, the circle has a radius of $\\sqrt{7}$ and center $O$. Points $A$, $B$, and $C$ are on the... | 4 |
[
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"role": "user",
"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 total number of integer solutions of the equation $x^5 - y^2 = 4$.\n\nRemember to put your answer... | 0 |
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the polynomials:\n\\[ P(x) = (x + \\sqrt{2})(x^2 - 2x + 2) \\]\n\\[ Q(x) = (x - \\sqrt{2})(x^2 + ... | 0 |
[
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"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhile taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the Col... | 2013 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move wi... | 0 |
[
{
"role": "user",
"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 $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the plac... | 27 |
[
{
"role": "user",
"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 $5$ vertices labeled $1, 2, 3, 4, 5$. For any two pairs of vertices $u, v$, the edge $uv$ is dra... | 127 |
[
{
"role": "user",
"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 $r$ be a nonzero real number. The values of $z$ which satisfy the equation\n\\[ r^4z^4 + (10r^6-2r^2)z... | 8 |
[
{
"role": "user",
"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 the rational number $\\frac{p}{q}$ be closest to but not equal to $\\frac{22}{7}$ among all rational n... | 14 |
[
{
"role": "user",
"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 $x, y, z \\in \\mathbb{R}$ and the equation $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$ holds. Find the minimum v... | 7 |
[
{
"role": "user",
"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 a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest side has length $12$. What... | 3 |
[
{
"role": "user",
"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 real numbers $x$, let \n$$f(x) = \\frac{4^x}{25^{x+1}} + \\frac{5^x}{2^{x+1}}.$$ \nThen, \n$$f\\left(\... | 15001 |
[
{
"role": "user",
"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 remainder when $712!$ is divided by $719$.\n\nRemember to put your answer on its own line after \... | 718 |
[
{
"role": "user",
"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 the side lengths of triangle $ABC$ are the roots of the polynomial $x^3 - 27x^2 + 222x - 540$. Wha... | 10 |
[
{
"role": "user",
"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 each positive integer $n$, let $S_n$ denote the set of positive integers $k$ such that $n^k - 1$ is di... | 912 |
[
{
"role": "user",
"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\nHow many reorderings of \\(2, 3, 4, 5, 6\\) have the property that every pair of adjacent numbers are rela... | 4 |
[
{
"role": "user",
"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 some positive integers $m > n$, the quantities $a = \\text{lcm}(m, n)$ and $b = \\gcd(m, n)$ satisfy $... | 11 |
[
{
"role": "user",
"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 $x^3 - 3\\sqrt{3} x^2 + 9x - 3\\sqrt{3} - 64 = 0$, find the value of $x^6 - 8x^5 + 13x^4 - 5x^3 + 49x^2... | 1898 |
[
{
"role": "user",
"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 $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\\pi$ and... | 47 |
[
{
"role": "user",
"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 numbers in the sequence 101, 104, 109, 116, $\\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, ... | 401 |
[
{
"role": "user",
"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\nWhen $4 \\cos \\theta - 3 \\sin \\theta = \\frac{13}{3},$ it follows that $7 \\cos 2\\theta - 24 \\sin 2\\... | 122 |
[
{
"role": "user",
"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 $\\{a_n\\}$ satisfies the following conditions: \n- $a_0 = 0$ \n- $a_1 = a_2 = 1$ \n- $a_{3n... | 6 |
[
{
"role": "user",
"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\nEuler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + ... | 250 |
[
{
"role": "user",
"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(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \\(\\frac{1}{a^3 + b^3} + \\frac{1}{b^3 + c^3} ... | 3989 |
[
{
"role": "user",
"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 George Washington Bridge is $2016$ meters long. Sally is standing on the bridge, $1010$ meters from it... | 1016060 |
[
{
"role": "user",
"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 the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$... | 4648 |
[
{
"role": "user",
"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 $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer... | 1 |
[
{
"role": "user",
"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$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$. The cubic polyn... | 2015000 |
[
{
"role": "user",
"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 number of solutions to the equation: \n\n$$6\\{x\\}^3 + \\{x\\}^2 + \\{x\\} + 2x = 2018.$$ \n\nWi... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the fir... | 93 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of ... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straig... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x$.\n\nRemember... | 8 |
[
{
"role": "user",
"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\nBroady The Boar is playing a board game consisting of a circle with $2021$ points on it, labeled $0$, $1$,... | 2021 |
[
{
"role": "user",
"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 greatest positive integer $n$ such that $3^n$ divides $70! + 71! + 72!$.\n\nRemember to put your ... | 36 |
[
{
"role": "user",
"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\nDetermine the largest natural number $m$ such that for each non-negative real numbers $a_1 \\ge a_2 \\ge \... | 44 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCounting from the right end, what is the $2500$th digit of $10000!$?\n\nRemember to put your answer on its... | 2 |
[
{
"role": "user",
"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\nDetermine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$.\n\nRemember to put your answer on its own line ... | 9 |
[
{
"role": "user",
"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\nWhile there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there... | 132 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last three digits of \n\\[2008^{2007^{\\cdot^{\\cdot^{\\cdot ^{2^1}}}}}.\\]\n\nRemember to put yo... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, W... | 71622400 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree non-zero real numbers $a$, $b$, and $c$ satisfy the conditions $a + b + c = 0$ and $a^4 + b^4 + c^4 ... | -8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$ be positive real numbers for which:\n\\[\n \\frac{5}{a} = b+c, \\quad\n \\frac{10}{b} ... | 55 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are five dots arranged in a line from left to right. Each of the dots is colored from one of five co... | 2800 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn acute triangle $ABC$ has side lengths $a$, $b$, $c$ such that $a$, $b$, $c$ form an arithmetic sequence... | 42 |
[
{
"role": "user",
"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 three-digit number $999$ has a special property: It is divisible by $27$, and its digit sum is also di... | 75 |
[
{
"role": "user",
"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 $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\\tri... | 251 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered triples of integers $(x, y, z)$ are there such that $x^2 + y^2 + z^2 = 34$?\n\nRemember t... | 48 |
[
{
"role": "user",
"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 bijective function $g : \\mathbb{R} \\to \\mathbb{R}$, we say that a function $f : \\mathbb{R} \\to ... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $\\triangle ABC$ has $\\angle{A}=90^\\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is... | 149 |
[
{
"role": "user",
"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\nAt a recent math contest, Evan was asked to find $2^{2016} \\pmod{p}$ for a given prime number $p$ with $1... | 211 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFarmer James invents a new currency such that for every positive integer $n \\leq 6$, there exists an $n$-... | 210 |
[
{
"role": "user",
"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 board with dimensions $2018 \\times 2018$ is divided into unit cells of size $1 \\times 1$. In some cell... | 1018081 |
[
{
"role": "user",
"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 how many integer values $3 \\le n \\le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n}... | 49 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA chicken breeder went to check the price per chick he charged the previous year. He found an invoice, hal... | 511 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n$ is a perfect number and $\\varphi(n)$ is a power of $2$.\n\nR... | 6 |
[
{
"role": "user",
"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 $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\o... | 3225 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three... | 111 |
[
{
"role": "user",
"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 trapezoid has side lengths as indicated in the figure, with the sides of lengths $11$ and $36$ being par... | 564 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour friends purchased a ball. The first friend paid half of the ball's total price. The second friend con... | 100 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last two digits of $\\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e... | 20 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the minimum value of \\( S = |x + 1| + |x + 5| + |x + 14| + |x + 97| + |x + 1920| \\).\n\nRemember to... | 2011 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind \\( \\sum_{i=1}^{2016} i(i+1)(i+2) \\pmod{2018} \\).\n\nRemember to put your answer on its own line a... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest positive integer that is a divisor of all three of \\( A=2^{2010}\\times3^{2010} \\)... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ such that $AP = 4$ and $BP = 3$.... | 90 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\((2^x - 4^x) + (2^{-x} - 4^{-x}) = 3\\), find the numerical value of the expression \n\\[\n(8^x + 3\\... | -1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBeto plays the following game with his computer: initially, the computer randomly picks $30$ integers from... | 11 |
[
{
"role": "user",
"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 $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible v... | 112 |
[
{
"role": "user",
"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\nXavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equa... | 1006 |
[
{
"role": "user",
"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 lines $m$, $n$, and $\\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at ... | 40 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many sequences of words (not necessarily grammatically correct) have the property that the first word ... | 60 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers... | 105 |
[
{
"role": "user",
"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 quadrilateral $ABCD$ is an isosceles trapezoid with $AB = CD = 1$, $BC = 2$, and $DA = 1 + \\sqrt{3}$.... | 90 |
[
{
"role": "user",
"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 $pq$ for all pairs of prime numbers $p \\leq q$ such that $pq \\mid 15(p-1)(q-1)$.\n\nReme... | 31 |
[
{
"role": "user",
"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 least positive integer $n$ such that there exists a polynomial \\[ P(x) = a_{2n}x^{2n} + a_{2n-1}... | 2014 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag fo... | 45 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2... | 141 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest integer whose prime factors add to $14$?\n\nRemember to put your answer on its own lin... | 162 |
[
{
"role": "user",
"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 number with $n$ digits ($n > 1$). A number $b$ with $2n$ digits is obtained by writing two co... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape fro... | 1278 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest $n$ for which there exists a sequence $(a_0, a_1, \\ldots, a_n)$ of non-zero digits such ... | 4 |
[
{
"role": "user",
"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 the number of two-digit numbers $\\overline{ab}$ with nonzero digits $a$ and $b$ such that $a$ and... | 14 |
[
{
"role": "user",
"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 first $55$ terms of the sequence $\\binom{0}{0}, \\binom{1}{0}, \\binom{1}{1}, \\binom... | 1023 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of integers of the form $2^x + 2^y + 2^z$, where $x$, $y$, $z$ are pairwise distinct no... | 577 |
[
{
"role": "user",
"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 alphabet in its natural order $\\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to... | 252 |
[
{
"role": "user",
"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 number of integers $n$ such that $1 \\leq n \\leq 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple ... | 99 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIt is given that there exists a unique triple of positive primes \\((p, q, r)\\) such that \\(p < q < r\\)... | 19 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to ... | 50 |
[
{
"role": "user",
"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 decimal digits of \\[ \\left\\lfloor \\frac{51525354555657\\dots979899}{50} \\right\\r... | 457 |
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