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\nLet $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line $AP$, p... | 20 |
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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 $Answer is the answer to the problem.\n\nSolve the equation:\n\\[\n \\sin 9^\\circ \\sin 21^\\circ \\sin(102^\\circ + x^\\circ) = \\sin 30^\\circ \\sin 42^\\circ \\si... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice, where each drink has a pos... | 7 |
[
{
"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 an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Find the la... | 47 |
[
{
"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 triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, where $y < ... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square has been divided into $2022$ rectangles, with no two of them having a common interior point. What is the maximal numb... | 2025 |
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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\nInitially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgrap... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line as... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $m$ is called perfect if the sum of all its positive divisors, including $1$ and $m$, equals $2m$. Determin... | 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\nGiven the function $f(x) = x^3 - 6x^2 + 17x$, if $f(a) = 16$ and $f(b) = 20$, find the value of $a + b$.\n\nRemember to put yo... | 4 |
[
{
"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 circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane inter... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA table tennis club hosts a series of doubles matches following several rules:\n1. Each player belongs to two pairs at most.\n... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center i... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive i... | 5 |
[
{
"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 each positive integer $n$, assign a non-negative integer $f(n)$ such that the following conditions are satisfied:\n\n1. $f(... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all the ways of placing the integers $1, 2, 3, \\ldots, 16$ in the boxes below, such that each integer appears in exactly... | 1 |
[
{
"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\nSeven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at le... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest $k$ such that for every positive integer $n$, there exist at least $k$ numbers in the set \\( \\{ n+1, n+2, \... | 1 |
[
{
"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 are the last two digits of $2^{3^{4^{\\cdots^{2019}}}}$?\n\nRemember to put your answer on its own line after \"Answer:\"... | 52 |
[
{
"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\nPositive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2 -... | 64 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers \\((p, q)\\) satisfying the equation \\(p^2 - q^2 + p + q = 2014\\).\n\nRemember ... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe call a pair $(a,b)$ of positive integers a \"pupusa\" if $a < 391$ and \\(\\text{lcm}(a,b) > \\text{lcm}(a,391)\\). Find th... | 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\nLet $x$ be a real number in the interval $(0, \\frac{\\pi}{2})$ such that $\\frac{1}{\\sin x \\cos x} + 2\\cot 2x = \\frac{1}{... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$.\n\nRemember to put your an... | 2 |
[
{
"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 multiple of $2019$ that is of the form $abcabc\\dots abc$, where $a$, $b$, and $c$ are digits.\n\nRemember t... | 673673673 |
[
{
"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 sum of all possible values of $\\cos(2\\theta)$ if $\\cos(2\\theta) = 2\\cos(\\theta)$ for a real number $\\theta$... | 2 |
[
{
"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 $AB=4$, $AC=5$, $BC=6$, and circumcircle $\\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respe... | 1763 |
[
{
"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 $A$ is adjacent to square $B$, which is adjacent to square $C$. The three squares all have their bottom sides along a c... | 54 |
[
{
"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\nProblem 1: The set $S = \\{1, 2, 3, \\ldots, 2006\\}$ is partitioned into two disjoint subsets $A$ and $B$ such that:\n\n1. $1... | 154 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $2024^{2023^{2022^{2021 \\ldots^{3^{2}}}}} + 2025^{2021^{2017^{2013 \\ldots^{5^{1}}}}}$ is divided by ... | 17 |
[
{
"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 an isosceles triangle $ABC$ with side lengths $AB = AC = 10\\sqrt{2}$ and $BC = 10\\sqrt{3}$. Construct semicircles $... | 24 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.\n\nRemember to put your answer on i... | 5 |
[
{
"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 number of permutations of the set $\\{1, 2, 3, \\ldots, 2014\\}$ such that for every two different numbers $a... | 2014 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = 1 + x + x^2 + \\cdots + x^{100}$. Find $f'(1)$.\n\nRemember to put your answer on its own line after \"Answer:\"."... | 5050 |
[
{
"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 integers $x_1, x_2, \\ldots$ is called \"double-dipped\" if it satisfies the recurrence relation $x_{n+2} = ax_{... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following con... | 315 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are... | 160 |
[
{
"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 positive integer $n$ such that $n^2$ is the perfect square closest to the sum $8 + 16 + 24 + \\cdots + 8040$.\n\nReme... | 2011 |
[
{
"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 ordered quadruples $(a,b,c,d)$ of integers, where $a,b,c,d$ are integers between $1$ and $3$ inclusive... | 66 |
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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 the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \\[ \\frac{1}... | 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 $Answer is the answer to the problem.\n\nThe trapezoid below has bases with lengths 7 and 17, and an area of 120. Find the difference in the areas of the two triangles... | 50 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe say that an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a + b$ and $a - b$ are perfect sq... | 16 |
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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 $\\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$. Similarl... | 120 |
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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 spider is walking on the boundary of an equilateral triangle $\\triangle{ABC}$, with vertices labeled in counterclockwise or... | 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\nCompute the sum of the digits of $101^6$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "use... | 28 |
[
{
"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 polynomial $f(x) = ax^2 + bx + c$, with degree less than or equal to 2.\n\nWhen $f$ varies subject to the constra... | 2 |
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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 $Answer is the answer to the problem.\n\nFive girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at l... | 118 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum nu... | 2 |
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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 $\\ell$ be a line and let points $A$, $B$, $C$ lie on $\\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in oppos... | 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\nIn a rectangular $57 \\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ ... | 457 |
[
{
"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 right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\\text{lcm}(a,b)=72$, $\\text{lcm}(a,c)=24$, and $\\... | 3024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \\le n$, define \n\\[ S_m = \\sum_{\\substack{i-j=m \\\\ ... | 8027 |
[
{
"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 smallest subset of the integers with the property that $0 \\in S$ and for any $x \\in S$, we have $3x \\in S$ a... | 128 |
[
{
"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 $\\overline{ABCD}$ be a 4-digit number. What is the smallest possible positive value of $\\overline{ABCD} - \\overline{DCB... | 90 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $r_k$ denote the remainder when $\\binom{127}{k}$ is divided by $8$. Compute $r_1 + 2r_2 + 3r_3 + \\cdots + 63r_{63}$.\n\n... | 2016 |
[
{
"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\nGiven a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\\lceil \\frac{n}{2} \\rc... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe following sequence lists all the positive rational numbers that do not exceed $\\frac{1}{2}$ by first listing the fraction... | 61 |
[
{
"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 function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2)=0$, $f(3... | 660 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $M = \\{1,2,\\cdots , 10\\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\\{a,b\\},... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the value of $x$ where the graph of \\( y = \\log_3(\\sqrt{x^2+729}+x) - 2\\log_3(\\sqrt{x^2+729}-x) \\) crosses the $x$-... | 36 |
[
{
"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 integers such that $a + 2b + 3c = 100$. Find the greatest value of $M = abc$.\n\nRemember to put you... | 6171 |
[
{
"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 positive integer $n$ such that:\n\\[\n\\underbrace{f(f(\\cdots f}_{2013 \\ f\\text{'s}}(n)\\cdots ))=2014^2+1\n\\]\nw... | 1015057 |
[
{
"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 positive integer $n > 1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n) = n - g(n)$. For exampl... | 19 |
[
{
"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\nAltitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line ... | 33725 |
[
{
"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 sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j... | 203 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, a... | 47 |
[
{
"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 sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \\frac{2\\Delta}{x} = y + \\frac{2\\Delta}{y}$, where $\... | 87 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine the function $f: \\mathbb{R} \\setminus \\{-1,1\\} \\to \\mathbb{R}$ as follows:\n\\[ f(x) = \\sum_{a,b=0}^{\\infty} \\... | 4033 |
[
{
"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\nGiven a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where each coefficient $a, b, c, d, e,$ and $f$ is either $1$ or... | 142 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, \\( \\overline{AB} \\) and \\( \\overline{CD} \\) are parallel, \\( \\angle BXY = 45^\\circ \\), \\( \\a... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^{2} + ax + b$. If for all nonzero real $x$:\n\\[ f\\left(x + \\frac{1}{x}\\right) = f\\left(x\\right) + f\\left(... | 13 |
[
{
"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\nGiven a triangle $ABC$ with angles $\\angle A = 60^{\\circ}$, $\\angle B = 75^{\\circ}$, $\\angle C = 45^{\\circ}$, let $H$ be... | 1132 |
[
{
"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 know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\\triangle ... | 622 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the re... | 166464 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a circle, 15 equally spaced points are drawn, and arbitrary triangles are formed by connecting 3 of these points. How many ... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGraphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the ma... | 1005 |
[
{
"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\nEvaluate $2023 \\cdot \\frac{2023^6 + 27}{(2023^2 + 3)(2024^3 - 1)} - 2023^2$.\n\nRemember to put your answer on its own line ... | -6066 |
[
{
"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 $x, y$ be positive real numbers. If \\[129-x^2=195-y^2=xy,\\] then $x = \\frac{m}{n}$ for relatively prime positive intege... | 4306 |
[
{
"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 rectangular strips, each measuring $4$ by $16$ inches, are laid out with two vertical strips crossing two horizontal stri... | 96 |
[
{
"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 each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in eac... | 9 |
[
{
"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 flipped, coin A shows heads $\\frac{1}{3}$ of the time, coin B shows heads $\\frac{1}{2}$ of the time, and coin C shows h... | 273 |
[
{
"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 wish to find the sum of $40$ given numbers using $40$ processors. Initially, each processor displays the number $0$ on its ... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf\n\\[\n\\sum_{k=1}^{1000}\\left( \\frac{k+1}{k}+\\frac{k}{k+1}\\right)=\\frac{m}{n}\n\\]\nfor relatively prime positive inte... | 2004001 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two... | 2128 |
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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 $Answer is the answer to the problem.\n\nFind the decimal form of the largest prime divisor of \\(100111011_6\\).\n\nRemember to put your answer on its own line after ... | 181 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5... | 1010 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any interval $\\mathcal{A}$ on the real number line not containing zero, define its reciprocal to be the set of numbers of... | 60 |
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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 $n$ be an arbitrary positive integer. Calculate \\( S_n = \\sum_{r=0}^n 2^{r-2n} \\binom{2n-r}{n} \\).\n\nRemember to put ... | 1 |
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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 what single digit $n$ does $91$ divide the 9-digit number $12345n789$?\n\nRemember to put your answer on its own line afte... | 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\nIn trapezoid $ABCD$, $AB < CD$, $AB \\perp BC$, $AB \\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P... | 61 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $24$-hour digital clock shows times $h : m : s$, where $h$, $m$, and $s$ are integers with $0 \\leq h \\leq 23$, $0 \\leq m ... | 1164 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that $f(x + y) = f(x) + f(y) + f(xy)$ for all $x, y \\in \\mathbb{R}... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a positive integer $N \\ge 2$ \"special\" if for every $k$ such that $2 \\leq k \\leq N$, $N$ can be expressed as a sum o... | 50 |
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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 $(a_n)_{n \\geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $m$ such that $529^n + m \\cdot 132^n$ is divisible by $262417$ for all odd positive i... | 1984 |
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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 area of the region in the $xy$-plane satisfying the inequality \n\\[\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4... | 210 |
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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 integers from $1$ through $9$ inclusive are placed in the squares of a $3 \\times 3$ grid. Each square contains a differen... | 17 |
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