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 $Answer is the answer to the problem.\n\nFind the minimum value of \\( \\sum_{i=1}^{10} \\sum_{j=1}^{10} \\sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)| \\), wher... | 2394000000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAngeline starts with a 6-digit number and moves the last digit to the front. For example, if she originally had $100823$, she ... | 102564 |
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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 $n$ such that $2^{2000}$ divides $n!$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 2008 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1 \\times... | 61 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the maximal possible length of the sequence of consecutive integers which can be expressed in the form $x^3 + 2y^2$,... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each other. Th... | 10 |
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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 $Answer is the answer to the problem.\n\nTim is participating in the following three math contests. On each contest, his score is the number of correct answers.\n- The... | 50 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( n \\) be a non-negative integer. Define the *decimal digit product* \\( D(n) \\) inductively as follows:\n\n- If \\( n... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe operation $\\ast$ is defined by $a \\ast b = a + b + ab$, where $a$ and $b$ are real numbers. Find the value of:\n\n\\[\n\... | 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 least positive integer $n$ such that $\\cos\\frac{\\pi}{n}$ cannot be written in the form $p+\\sqrt{q}+\\sqrt[3]{r}$ ... | 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 $Answer is the answer to the problem.\n\nDefine the maximal value of the ratio of a three-digit number to the sum of its digits. Let the three-digit number be represen... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is d... | 1090 |
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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 vertices of $\\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with tw... | 344 |
[
{
"content": "Solve the following math problem step by 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 Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\\ge 2$. Find the... | 256 |
[
{
"content": "Solve the following math problem step by 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 regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. Th... | 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\nThe numbers $1, 2, \\ldots, 33$ are written on the board. A student performs the following procedure: choose two numbers from ... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last two digits of $2023 + 202^3 + 20^{23}$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 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\nLet the sequence $a_{1}, a_{2}, \\ldots$ be defined recursively as follows: $a_{n} = 11a_{n-1} - n$. If all terms of the seque... | 121 |
[
{
"content": "Solve the following math problem step by 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 integer $n \\ge 1$, define $a_n = \\left\\lfloor \\frac{n}{\\left\\lfloor \\sqrt{n} \\right\\rfloor} \\right\\rfloor$... | 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\nFind the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10} + 2133$ is a perfect cube. \n\nSolve th... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a board, the numbers $1, 2, 3, \\ldots, 98, 99$ are written. One must mark $50$ of them such that the sum of any two marked... | 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 $Answer is the answer to the problem.\n\nNine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of pr... | 280 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $x = \\cos 1^\\circ \\cos 2^\\circ \\cos 3^\\circ \\ldots \\cos 89^\\circ$ and $y = \\cos 2^\\circ \\cos 6^\\circ \\cos 10^... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAl told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct posit... | 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\nIn the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a const... | 1012 |
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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 $Answer is the answer to the problem.\n\nSay that a 4-digit positive integer is \\textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest ... | 1680 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of different numbers of the form $\\left\\lfloor\\frac{i^2}{2015} \\right\\rfloor$, with $i = 1, 2, \\ldots, 2... | 2016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a right triangle with $\\angle ACB = 90^{\\circ}$. $D$ is a point on $AB$ such that $CD \\perp AB$. If the area o... | 1008 |
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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\\textbf{Problem 5.} Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $\\frac{15}{10}$ minu... | 1440 |
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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\nYukihira is counting the minimum number of lines $m$ that can be drawn on the plane so that they intersect in exactly $200$ di... | 21 |
[
{
"content": "Solve the following math problem step by 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 $n$ with $1 \\leq n \\leq 100$ for which $n - \\phi(n)$ is prime. Here, $\\phi(n)$ denotes the num... | 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\nFind the integer closest to the value of the following expression:\n\n$$\\left((3 + \\sqrt{1})^{2023} - \\left(\\frac{1}{3 - \... | 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\nLet $O$ and $A$ be two points in the plane with $OA = 30$, and let $\\Gamma$ be a circle with center $O$ and radius $r$. Suppo... | 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\nConsider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 +... | 169 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the numbe... | 118 |
[
{
"content": "Solve the following math problem step by 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 $\\frac{1}{1}, \\frac{1}{2}, \\ldots, \\frac{1}{2010}$ are written on a blackboard. A student chooses any two of t... | 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\nMarko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\\frac{2}{9}$, $1$ u... | 1811716 |
[
{
"content": "Solve the following math problem step by 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 group of \\( n \\) friends participated in a math contest consisting of eight short-answer problems \\( S_1, S_2, S_3, S_4, ... | 10 |
[
{
"content": "Solve the following math problem step by 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 exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, ... | 2001 |
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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 people in an infinitely long line are numbered $1,2,3,\\dots$. Then, each person says either \"Karl\" or \"Lark\" independ... | 1028 |
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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 4-digit numbers with distinct digits chosen from the set $\\{0,1,2,3,4,5\\}$ in which no two adjacent digit... | 150 |
[
{
"content": "Solve the following math problem step by 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$ which is both a multiple of 19 and whose digits add to 23.\n\nRemember to put your answer ... | 779 |
[
{
"content": "Solve the following math problem step by 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_0, a_1, a_2, \\dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each po... | 1332 |
[
{
"content": "Solve the following math problem step by 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 \\emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of si... | 243 |
[
{
"content": "Solve the following math problem step by 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 diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle ... | 640 |
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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 subset $S$ of the set of integers $0$ to $99$ is said to have property $A$ if it is impossible to fill a crossword-puzzle wi... | 25 |
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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 a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so t... | 365 |
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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\nWhat is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?\n\nRemember to p... | 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\nDefine $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \\cdots + a_k$. Let \\[\\frac{a_{100}}{b_{100}} = \\frac{m}{n}\\] wher... | 99 |
[
{
"content": "Solve the following math problem step by 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 $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \\leq k \\leq... | 32 |
[
{
"content": "Solve the following math problem step by 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 period of the repetend of the fraction $\\frac{39}{1428}$ by using binary numbers, i.e., its binary decimal represent... | 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 $x$ be a real number such that:\n\\[ 4^{2x} + 2^{-x} + 1 = (129 + 8\\sqrt{2})(4^{x} + 2^{-x} - 2^{x}). \\]\nFind $10x$.\n\... | 35 |
[
{
"content": "Solve the following math problem step by 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 of integers $a_1, a_2, \\dots, a_k$ where every term $a_i \\in \\{1,2\\}$, and let $S$ denote their sum. Ano... | 476 |
[
{
"content": "Solve the following math problem step by 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 well-known that the $n^{\\text{th}}$ triangular number can be given by the formula $\\frac{n(n+1)}{2}$. A Pythagorean tr... | 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\nA week ago, Sandy's seasonal Little League batting average was $360$. After five more at bats this week, Sandy's batting avera... | 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\nLet \\( G, O, D, I, \\) and \\( T \\) be digits that satisfy the following equation:\n\n\\[\n\\begin{array}{cccc}\n & G & O & ... | 10908 |
[
{
"content": "Solve the following math problem step by 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 classroom has $30$ students, each of whom is either male or female. For every student $S$, we define their ratio to be the n... | 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\nThere are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\\bino... | 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\nIn equilateral triangle $ABC$, the midpoint of $\\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $... | 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 $T$ denote the 15-element set $\\{10a+b : a,b \\in \\mathbb{Z}, 1 \\le a < b \\le 6\\}$. Let $S$ be a subset of $T$ in whi... | 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\nLet $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible val... | 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\nFind the smallest constant $c$ such that for every 4 points in a unit square, there are two points with a distance less than o... | 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\nSherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between ... | 81 |
[
{
"content": "Solve the following math problem step by 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 $80 \\times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square... | 40 |
[
{
"content": "Solve the following math problem step by 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 incircle $\\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius ... | 74 |
[
{
"content": "Solve the following math problem step by 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 faces of a cube contain the numbers 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7.... | 343 |
[
{
"content": "Solve the following math problem step by 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_1, A_2, \\ldots, A_n$ an $n$-division of $A$ if:\n\n1. $A_1 \\cap A_2 \\cap \\cdots \\cap A_n = A$,\n2. $A_i \\cap ... | 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\nThere are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we ta... | 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\nConsider all ordered pairs \\((m, n)\\) of positive integers satisfying the equation \\(59m - 68n = mn\\). Find the sum of all... | 237 |
[
{
"content": "Solve the following math problem step by 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 a set of $2020$ distinct points in the plane. Define \n\\[M=\\{P : P \\text{ is the midpoint of } XY \\text{ for so... | 4037 |
[
{
"content": "Solve the following math problem step by 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 maximum possible value of \\[ \\frac{\\left(x^2+5x+12\\right)\\left(x^2+5x-12\\right)\\left(x^2-5x+12\\right)\\l... | 576 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are playing a game called \"Hovse.\" Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ ... | 127 |
[
{
"content": "Solve the following math problem step by 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 positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one... | 2636 |
[
{
"content": "Solve the following math problem step by 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 square number with 4 digits, such that all its digits are less than 6. If we add 1 to each digit, the resulting n... | 2025 |
[
{
"content": "Solve the following math problem step by 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 $a^{2005} + b^{2005}$ can be expressed as a polynomial of $a + b$ and $ab$. Find the sum of the coefficients ... | 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\nLet $ABC$ be an isosceles triangle with $\\angle A = 90^\\circ$. Points $D$ and $E$ are selected on sides $AB$ and $AC$, and p... | 100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsing each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine... | 247 |
[
{
"content": "Solve the following math problem step by 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$ be a prime number such that $\\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number of divis... | 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\nConsider a triangle $ABC$ with $AB = 13$, $BC = 14$, $CA = 15$. A line perpendicular to $BC$ divides the interior of $\\triang... | 112 |
[
{
"content": "Solve the following math problem step by 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 sweeties shop called \"Olympiad\" sells boxes of $6$, $9$, or $20$ chocolates. Groups of students from a nearby school col... | 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\nFind the number of ordered triples of positive integers \\((a, b, c)\\) such that \\(abc\\) divides \\((ab + 1)(bc + 1)(ca + 1... | 16 |
[
{
"content": "Solve the following math problem step by 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$ is a real number such that $3^x = 27x$, compute $\\log_3 \\left(\\frac{3^{3^x}}{x^{3^3}}\\right)$.\n\nRemember to put y... | 81 |
[
{
"content": "Solve the following math problem step by 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 cross is the figure composed of $6$ unit squares as shown below (and any figure made of it by rotation).\n\n![Cross Image](h... | 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\nSuppose that $a$ and $b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ ... | 296 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ be a positive real number and $n$ a non-negative integer. Determine $S-T$, where \n$$ S = \\sum_{k=-2n}^{2n+1} \\frac{... | 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 $Answer is the answer to the problem.\n\nLil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A=(a_{ij})$ be the $n \\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTo each element of the set $S = \\{1, 2, \\ldots, 1000\\}$ a color is assigned. Suppose that for any two elements $a, b$ of $S... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a fri... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHenry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Hen... | 904 |
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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\nDefine $f(n) = \\frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \\leq 1000$ and $f(n)$ is the ... | 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\nYou roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the thre... | 706 |
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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 $Answer is the answer to the problem.\n\n$A$ and $B$ play a game where $A$ chooses a positive integer $n \\in \\{1, 2, \\dots, 1001\\} = S$. $B$ must guess the value o... | 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\nGiven in the figure:\n- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.\n- The area of the t... | 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\nFind the least integer $n$ greater than $345$ such that $\\frac{3n+4}{5}$, $\\frac{4n+5}{3}$, and $\\frac{5n+3}{4}$ are all in... | 397 |
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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 following: $$\\sum^{99}_{x=0} (x^2 + 1)^{-1} \\pmod{199}$$ where $x^{-1}$ is the value $0 \\le y \\le 199$ such th... | 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\nA frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only al... | 256 |
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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 largest integer $n$ for which $2^n$ divides \\[ \\binom{2}{1} \\binom{4}{2} \\binom{6}{3} \\dots \\binom{128}{64}. \\... | 193 |
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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 triangle $ABC$, consider the semicircle with diameter $\\overline{EF}$ on $\\overline{BC}$ tangent to $\\overline{AB}$... | 84 |
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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 $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ ... | 3954 |
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