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\nIn a French village, the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people w... | 2401 |
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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 a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, ... | 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 $Answer is the answer to the problem.\n\nDetermine the remainder when \\((x^4-1)(x^2-1)\\) is divided by \\(1+x+x^2\\).\n\nRemember to put your answer on its own line ... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron... | 270 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circular disc is divided into $12$ equal sectors, each of which is colored using one of $6$ different colors. No two adjacen... | 20346485 |
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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 $\\mathcal{A}$ be the set of finite sequences of positive integers $a_1, a_2, \\dots, a_k$ such that $|a_n - a_{n-1}| = a_... | 1597 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. I... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting fro... | 182 |
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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\nHow many positive integers divide at least two of the numbers $120$, $144$, and $180$?\n\nRemember to put your answer on its o... | 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\nFor $k \\in \\{ 0, 1, \\ldots, 9 \\}$, let $\\epsilon_k \\in \\{-1, 1 \\}$. If the minimum possible value of $\\sum_{i = 1}^9 ... | 174762 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \\le n \\le 1000$ such that if $a_0 =... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.\n\nRemember to put your answer o... | 760 |
[
{
"content": "Solve the following math problem step by 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 ant lies on each corner of a $20 \\times 23$ rectangle. Each second, each ant independently and randomly chooses to move on... | 130 |
[
{
"content": "Solve the following math problem step by 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 four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it vanishes,... | 24 |
[
{
"content": "Solve the following math problem step by 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 $a_1$, $a_2$, \\ldots, $a_6$; $b_1$, $b_2$, \\ldots, $b_6$; and $c_1$, $c_2$, \\ldots, $c_6$ be permutations of $1$, $2$, ... | 162 |
[
{
"content": "Solve the following math problem step by 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 $m, n \\in \\mathbb{N}^*$. Find the least $n$ for which there exists $m$, such that the rectangle $(3m + 2) \\times (4m + ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRob is helping to build the set for a school play. He needs to construct a multi-colored tetrahedron using cloth and bamboo. T... | 65 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n \\geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.\n\nRemember to ... | 53 |
[
{
"content": "Solve the following math problem step by 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 $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ such that $\\overline{B... | 1245 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that \\( p(x) \\) is a polynomial and that \\( p(x) - p'(x) = x^2 + 2x + 1 \\). Compute \\( p(5) \\).\n\nRemember to p... | 50 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJesse has ten squares, labeled $1, 2, \\dots, 10$. In how many ways can he color each square either red, green, yellow, or blu... | 2304 |
[
{
"content": "Solve the following math problem step by 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 a polygon with $2013$ sides labeled as $A_{1}A_{2}\\ldots A_{2013}$. The vertices are marked with numbers such tha... | 67 |
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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\nIt is known that odd prime numbers $x, y, z$ satisfy the following conditions:\n\n- $x \\mid (y^5 + 1)$\n- $y \\mid (z^5 + 1)$... | 2013 |
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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 highest degree $k$ of $1991$ for which $1991^k$ divides the number \n\\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}. \\]... | 1991 |
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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 smallest positive integer $n$ such that the 73 fractions $\\frac{19}{n+21}, \\frac{20}{n+22}, \\frac{21}{n+23}, \\ldo... | 95 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k, m \\in \\mathbb{N} = \... | 79 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers $0 \\le a \\le 124$ such that $a^3 - 2$ is a multiple of $125$.\n\nRemember to put your answer on... | 53 |
[
{
"content": "Solve the following math problem step by 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 possible value of $\\frac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positiv... | 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\nThere are $2016$ customers who entered a shop on a particular day. Each customer entered the shop exactly once (i.e., the cust... | 45 |
[
{
"content": "Solve the following math problem step by 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 pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:\n\n1. $a*(b*c) = (a*b)*c$ for all... | 7776 |
[
{
"content": "Solve the following math problem step by 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 set $E = \\{5, 6, 7, 8, 9\\}$. For any partition $\\{A, B\\}$ of $E$, with both $A$ and $B$ non-empty, consider t... | 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\nLet $g(x) = 1 + 2x + 3x^2 + 4x^3 + \\ldots$. Find the coefficient of $x^{2015}$ in $f(x) = \\frac{g(x)}{1-x}$.\n\nRemember to ... | 2033136 |
[
{
"content": "Solve the following math problem step by 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 $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\\{ 1, 2, 3, \\dots, 17 \\}$, satisfy the equa... | 408 |
[
{
"content": "Solve the following math problem step by 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 triangle $ABC$, we have $\\angle ABC = 100^\\circ$, $\\angle ACB = 65^\\circ$, $M \\in AB$, $N \\in AC$, and $\\angle M... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that\n\\[\\sum_{x=1}^{70} \\sum_{y=1}^{70} \\frac{x^{y}}{y} = \\frac{m}{67!}\\] \nfor some positive integer $m$, find $m... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a $3 \\times 3$ board, the numbers from $1$ to $9$ are written in some order without repeating. We say that the arrangement... | 720 |
[
{
"content": "Solve the following math problem step by 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 an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ ... | 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\nFor a permutation $\\pi$ of the integers from 1 to 10, define\n\\[ S(\\pi) = \\sum_{i=1}^{9} (\\pi(i) - \\pi(i+1))\\cdot (4 + ... | 40320 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous being... | 1024 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a rectangle $ABCD$ such that $AB = 12$ and $BC = 7$. Points $E$ and $F$ lie on sides $AB$ and $CD$, respectively, suc... | 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\nIn a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cak... | 100 |
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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 a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some ai... | 2015 |
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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 positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \\cdots + a_k... | 95 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, wh... | 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\nAlex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more pr... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimete... | 41 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nShow that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number o... | 32 |
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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 $Answer is the answer to the problem.\n\nFind the numerator of \\(\\frac{1010\\overbrace{11 \\ldots 11}^{2011 \\text{ ones}}0101}{1100\\underbrace{11 \\ldots 11}_{2011... | 9091 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be an isosceles trapezoid with $AD \\parallel BC$ and $BC > AD$ such that the distance between the incenters of $\\... | 7114 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a rectangle $ABCD$, where the segment $MA$ is perpendicular to the plane $ABC$. The lengths are as follows: $MB = 15$, $... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor $a > 0$, find the minimum value of $\\int_{0}^{\\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\\ dx.$\n\nRemember to put your ans... | 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\nDefine a sequence $a_i$ as follows: \\( a_1 = 181 \\) and for \\( i \\ge 2 \\), \\( a_i = a_{i-1}^2 - 1 \\) if \\( a_{i-1} \\)... | 105 |
[
{
"content": "Solve the following math problem step by 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 $x$ satisfies $2 + \\log_{25} x + \\log_8 5 = 0$. Find $\\log_2 x - (\\log_8 5)^3 - (\\log_{25} x)^3$.\n\nRememb... | 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\nEach of the squares in a $2 \\times 2018$ grid is to be colored black or white such that in any $2 \\times 2$ block, at least ... | 1009 |
[
{
"content": "Solve the following math problem step by 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 quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the q... | 11 |
[
{
"content": "Solve the following math problem step by 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 non-zero natural number $n$ is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223 - \\frac{3}{2} \\... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed th... | 57 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe vertices of a regular nonagon are colored such that:\n1. Adjacent vertices are different colors.\n2. If 3 vertices form an... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$, $y$, and $z$ be real numbers such that:\n\\[\n12x - 9y^2 = 7 \\\\\n6y - 9z^2 = -2 \\\\\n12z - 9x^2 = 4\n\\]\nFind the... | 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 $Answer is the answer to the problem.\n\nThe figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven tri... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles with radius $2$ and radius $4$ have a common center at $P$. Points $A$, $B$, and $C$ on the larger circle are the ... | 192 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMark writes the expression $\\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression ... | 3480 |
[
{
"content": "Solve the following math problem step by 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 two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for ex... | 962963 |
[
{
"content": "Solve the following math problem step by 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 snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball ... | 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\nThere are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere ... | 2520 |
[
{
"content": "Solve the following math problem step by 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 ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at l... | 1248 |
[
{
"content": "Solve the following math problem step by 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 sixth digit after the decimal point in the number $(\\sqrt{1978} + \\lfloor \\sqrt{1978} \\rfloor)^{20}$.\n\nRem... | 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\nDistinct integers $x, y, z$ satisfy the equation $(x-y)(y-z)(z-x) = x+y+z$. Determine the smallest possible value of $|x+y+z|$... | 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\nLet the complex number $z = \\cos\\frac{1}{1000} + i \\sin\\frac{1}{1000}$. Find the smallest positive integer $n$ such that $... | 524 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the ... | 2013 |
[
{
"content": "Solve the following math problem step by 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, let $AB$ and $BC$ be chords, with $AB = \\sqrt{3}$, $BC = 3\\sqrt{3}$, and $\\angle ABC = 60^\\circ$. Find the le... | 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\nEvaluate $2000^3 - 1999 \\cdot 2000^2 - 1999^2 \\cdot 2000 + 1999^3$\n\nRemember to put your answer on its own line after \"An... | 3999 |
[
{
"content": "Solve the following math problem step by 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 whiteboard, the numbers are written sequentially: $1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \\ 8 \\ 9$. Andi has to paste a $+$ ... | 43 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $\\overline{a_1a_2\\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i = 1, 2, \\ldots, 2007$, the $2$-d... | 6 |
[
{
"content": "Solve the following math problem step by 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 $a$ and $b$ be two positive integers satisfying the equation:\n\\[ 20\\sqrt{12} = a\\sqrt{b}. \\]\nCompute the sum of all ... | 10800 |
[
{
"content": "Solve the following math problem step by 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 parallelogram $ABCD$ is $51\\sqrt{55}$ and $\\angle DAC$ is a right angle. If the side lengths of the parallelogra... | 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\nThe polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0$, $P_{1}(x)=x$, and \n\\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \\quad n\\... | 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\nIn the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with ... | 30 |
[
{
"content": "Solve the following math problem step by 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 integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero ... | 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\nDefine $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one in... | 2592 |
[
{
"content": "Solve the following math problem step by 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 natural number $k > 1$ is called \\textit{good} if there exist natural numbers \\(a_1 < a_2 < \\cdots < a_k\\) such that \\(... | 18 |
[
{
"content": "Solve the following math problem step by 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 every pair of real numbers $0 < a < b < 1$, we define sequences $\\{x_n\\}_{n \\ge 0}$ and $\\{y_n\\}_{n \\ge 0}$ by $x_0 ... | 307 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle. Point $P$ lies in the interior of $\\triangle ABC$ such that $\\angle ABP = 20^\\circ$ and $\\angle A... | 35 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest possible value of $x + y$ where $x, y \\ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.\n... | 11621 |
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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\nHow many integers between $2$ and $100$ inclusive cannot be written as $m \\cdot n$, where $m$ and $n$ have no common factors ... | 35 |
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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 $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining roo... | 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\nGiven that nonzero real numbers $a, b, c, d$ satisfy $a^b = c^d$ and $\\frac{a}{2c} = \\frac{b}{d} = 2$, compute $\\frac{1}{c}... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n**Problem 1.** The least common multiple of the numbers $a$ and $b$ is $88$. If $a^2 + b^2 = 2000$, find the value of $(a + b)... | 52 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a strictly increasing sequence $a_1 < a_2 < \\dots$ of positive integers such that each term $a_k$ is divisible either b... | 2010 |
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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 $\\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\\eta(\\eta(\\eta(10))) = 1... | 450 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \\{1, 3, 4\\} and ... | 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\nMarisa has two identical cubical dice labeled with the numbers $\\{1, 2, 3, 4, 5, 6\\}$. However, the two dice are not fair, m... | 28 |
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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\nCyclic quadrilateral $ABCD$ has $AC \\perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.... | 36 |
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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\nLaurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \\... | 28 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe real numbers $a_0, a_1, \\dots, a_{2013}$ and $b_0, b_1, \\dots, b_{2013}$ satisfy the recurrence relations:\n\\[\na_{n} =... | 671 |
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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 least positive integer that cannot be represented as $\\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digi... | 79 |
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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 positive integers $n$ for which $1^3 + 2^3 + \\cdots + 16^3 + 17^n$ is a perfect square.\n\nRemember to put your answ... | 3 |
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