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 triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point outside triangle $ABC$ suc... | 34 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter... | 113 |
[
{
"content": "Solve the following math problem step by 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 distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, a... | -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\nThere are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by o... | 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\nLet $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ \\sum_{(... | 37 |
[
{
"content": "Solve the following math problem step by 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 which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?\n\nRemember to put your answer on its own line after \"Answe... | 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\nFour positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such t... | 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 $Answer is the answer to the problem.\n\n$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How ... | 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\nDetermine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.\n\nRemember to put your ... | 509 |
[
{
"content": "Solve the following math problem step by 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 you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from ... | 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\nThere are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by on... | 1007 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the mi... | 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\nLet $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves t... | 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 $Answer is the answer to the problem.\n\nBurrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a squar... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDanial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and ea... | 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 $Answer is the answer to the problem.\n\nSabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDenote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$\n... | 402 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such th... | 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\nFind all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4... | 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\nA class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on ... | 300 |
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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 $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.\n\nRemember t... | 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 $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:\n\... | 37 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSome positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following... | -3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + ... | 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\nCharlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ c... | 8556 |
[
{
"content": "Solve the following math problem step by 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$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player ... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side l... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is ... | 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\nConsider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the mini... | 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\nDexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the pla... | 1010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $\\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ a... | 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\nFind the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfe... | 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 $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $... | 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\nPetya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pa... | 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\nDetermine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.\n\nRemember to put your answer on its o... | 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\nOn a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum ... | 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\nFor positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value ... | 1099 |
[
{
"content": "Solve the following math problem step by 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 $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquai... | 21 |
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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 ways can $8$ mutually non-attacking rooks be placed on a $9 \\times 9$ chessboard so that all $8$ rooks are on square... | 40320 |
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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 a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$... | 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 $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $B... | 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\nDetermine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.\n\nRemember to put your... | 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\nThree boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical whit... | 270901 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\\triangle ABC$ is... | 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 $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$... | 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\nFor any real numbers $x$ and $y$ that satisfy the equations:\n\\[\nx + y - xy = 155\n\\]\nand\n\\[\nx^2 + y^2 = 325\n\\]\nFind... | 4375 |
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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 $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly tw... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $k$ for which the equation:\n$$ \\text{lcm}(m,n) - \\text{gcd}(m,n) = k(m-n) $$\nhas no solution in... | 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\nDetermine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree a... | 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\nGiven a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ a... | 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\nFor some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of ... | 2021 |
[
{
"content": "Solve the following math problem step by 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 integers $a < b < c$ satisfy the following conditions:\n\\[\n a + b + c = 95 \\\\\n a^2 + b^2 + c^2 = 3083\n\\]\nFind ... | 37 |
[
{
"content": "Solve the following math problem step by 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 real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be ... | 132 |
[
{
"content": "Solve the following math problem step by 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 regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each ... | 1003 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nByan is playing a game called \"raven, raven, falcon\" with his three friends. His friends sit in a circle, and Byan walks clo... | 107 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular table unde... | 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\nOne day, students in school organized an exchange among themselves with the following rates:\n- $11$ strawberries exchange for... | 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\nIn triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extend... | 80 |
[
{
"content": "Solve the following math problem step by 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 integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.\n\nRemember to put your ans... | 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 the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.\n\nRemember to put yo... | 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\nThe numbers $1, 2, \\ldots, 10000$ are placed in the cells of a $100 \\times 100$ square grid, each number appearing exactly o... | 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\nThe triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of ... | 152 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the fi... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ re... | 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\nDetermine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural number.\n... | 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's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ ev... | 200 |
[
{
"content": "Solve the following math problem step by 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 $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, ... | 1987 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \\neq -f(-1)$. Suppose furthermore that the... | 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\nFind all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numb... | 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\nGiven that nonzero real numbers $x$ and $y$ satisfy the equations $x + \\frac{1}{y} = 3$ and $y + \\frac{1}{x} = 4$, find the ... | 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\nFind the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backwar... | 89 |
[
{
"content": "Solve the following math problem step by 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 values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficie... | 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\nFind the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 digits al... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet the number of ways for a rook to return to its original square on a $4 \\times 4$ chessboard in 8 moves, starting from a c... | 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 $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) is an isosceles triangle with \\( AB = AC \\). Point \\( P \\) is located inside \\( \\triangle ABC \\)... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two ne... | 720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this num... | 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 $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:\n\n1. $f(n)$ is a positive i... | 3816 |
[
{
"content": "Solve the following math problem step by 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 in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, ex... | 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\nReal nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:\n\\[\nA = \\frac{x}{y} + \\frac{y}{z} +... | -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\nFind all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\n\nRemember to put your answer on its o... | 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 sum of all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.\n\nRemember to pu... | 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\nLet $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:\n\\[\n\\begin{align*}\nx(y^2+yz+z^2)&=3y+10z... | 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\nGiven a pentagon $ABCDE$ with the following properties:\n\n1. Angles $\\angle A$ and $\\angle C$ are right angles.\n2. The sid... | 90 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.\n\nRemember ... | 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\nLet $\\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\\triangle ABC$ with its center on segment $BC$ su... | 415 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at point... | 936 |
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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\nPetya gave Vasya a number puzzle. Petya chose a digit $X$ and said, \"I am thinking of a three-digit number that is divisible ... | 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 $Answer is the answer to the problem.\n\nA Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to write down... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additionally, \... | 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\nWe are given a coin of diameter $\\frac{1}{2}$ and a checkerboard of $1 \\times 1$ squares with an area of $2010 \\times 2010$... | 6029 |
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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\nGraph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trin... | 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 $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:\n- $A'B' \\p... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, with $p, ... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsing each of the digits $1, 2, 3, \\ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Th... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest positive integer $n$ such that the following statement holds:\n\nIf $a_1, a_2, a_3, a_4, a_5, a_6$ are s... | 107 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSolve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3w + 9x +... | 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\nYou have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the... | 3025 |
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