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\nFor a positive integer $n$, let $f(n)$ be equal to $n$ if there exists an integer $x$ such that $x^2 - n$ is divisible by $2^{... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a right triangle with $AB=3$, $BC=4$, and $\\angle B = 90^\\circ$. Points $P$, $Q$, and $R$ are chosen on segment... | 451 |
[
{
"content": "Solve the following math problem step by 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 \\[\\left\\lfloor \\frac{2007!+2004!}{2006!+2005!}\\right\\rfloor.\\] (Note that $\\lfloor x \\rfloor$ denotes the gre... | 2006 |
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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 $Answer is the answer to the problem.\n\nFind the least three-digit number that is equal to the sum of its digits plus twice the product of its digits.\n\nRemember to ... | 397 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive i... | 161 |
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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 $p=101$. The sum \\[\\sum_{k=1}^{10}\\frac{1}{\\binom{p}{k}}\\] can be written as a fraction of the form $\\dfrac{a}{p!}$,... | 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\nEach cell of a $29 \\times 29$ table contains one of the integers $1, 2, 3, \\ldots, 29$, and each of these integers appears $... | 15 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvin and Jerry are playing a game with a pile of marbles. On each player's turn, they can remove $2$, $3$, $7$, or $8$ marbles... | 573 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssume the polynomial $p(x) = x^8 - 12x^6 + 49x^4 - 78x^2 + 42$ has no complex roots. How many negative roots does it have?\n\... | 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\nAndrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of po... | 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\nLet $a_0, a_1, \\ldots$ be a sequence of positive integers such that $a_0 = 1$, and for all positive integers $n$, $a_n$ is th... | 841 |
[
{
"content": "Solve the following math problem step by 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 real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in... | 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\nIf $x$, $y$, and $z$ are distinct positive integers such that $x^2 + y^2 = z^3$, what is the smallest possible value of $x + y... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the sum \\[1^2 + 2^2 - 3^2 - 4^2 + 5^2 + 6^2 - 7^2 - 8^2 + \\cdots - 1000^2 + 1001^2\\]\n\nRemember to put your answe... | 1001 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42$, and another is a... | 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\nWhat is the least positive integer $x$ for which the expression $x^2 + 3x + 9$ has 3 distinct prime divisors?\n\nRemember to p... | 27 |
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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 following diagram shows four adjacent $2 \\times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower l... | 49 |
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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 diagram below shows a circle with center $F$. The angles are related by the following equations:\n\n- $\\angle BFC = 2\\an... | 48 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn each cell of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:\n1. There are no two number... | 4032 |
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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 convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the di... | 396 |
[
{
"content": "Solve the following math problem step by 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 of all ordered $6$-tuples of nonnegative integers $(a, b, c, d, e, f)$ such that:\n\\[ a + 2b + 6c + 30d + 21... | 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\nThere are $n$ integers $a$ such that $0 \\leq a < 91$ and $a$ is a solution to the congruence $x^3 + 8x^2 - x + 83 \\equiv 0 \... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.\n\nRemember to put your a... | 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\nDefine a function $g : \\mathbb{N} \\rightarrow \\mathbb{R}$ such that:\n\\[\ng(x) = \\sqrt{4^x + \\sqrt{4^{x+1} + \\sqrt{4^{x... | 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\nLet $S = \\{ 1, 2, \\ldots, 15 \\}$. Let $A_1, A_2, \\ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions... | 15 |
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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 $Answer is the answer to the problem.\n\nThe Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of... | 1209 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe first $510$ positive integers are written on a blackboard: $1, 2, 3, \\ldots, 510$. An operation consists of erasing two n... | 255 |
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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 clueless ant makes the following route: starting at point $A$, it goes $1$ cm north, then $2$ cm east, then $3$ cm south, th... | 29 |
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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 number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.\n\nRememb... | 18 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine $a_k = 2^{2^{k-2013}} + k$ for all integers $k$. Simplify the expression:\n\n$$(a_0 + a_1)(a_1 - a_0)(a_2 - a_1)\\cdots... | 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\nA number is called \"cool\" if the sum of its digits is a multiple of $17$, and the sum of the digits of its successor is also... | 8899 |
[
{
"content": "Solve the following math problem step by 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 distinct positive integers $a, b < 2012$, define $f(a, b)$ to be the number of integers $k$ with $1 \\leq k < 2012$ such t... | 502 |
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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 sum of the non-repeated roots of the polynomial $P(x) = x^6 - 5x^5 - 4x^4 - 5x^3 + 8x^2 + 7x + 7$.\n\nRemember to put... | 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\nTwo bags contain some numbers, and the total number of numbers is prime.\n\nWhen we transfer the number 170 from bag 1 to bag ... | 167 |
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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\nIsosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of ... | 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\nA chess tournament is held with the participation of boys and girls. The number of girls is twice the number of boys. Each pla... | 33 |
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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$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the lin... | 33 |
[
{
"content": "Solve the following math problem step by 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) = \\sum_{d=1}^{n} \\left\\lfloor \\frac{n}{d} \\right\\rfloor$ and $g(n) = f(n) - f(n - 1)$. For how many $n$ from $... | 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\nA square grid $100 \\times 100$ is tiled in two ways: only with dominoes and only with $2 \\times 2$ squares. What is the mini... | 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\nWhat is the least possible sum of two positive integers $a$ and $b$ where $a \\cdot b = 10!$?\n\nRemember to put your answer o... | 3810 |
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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 $n$ schools in a city. The $i$th school dispatches $C_i$ students to watch a football match, where $1 \\leq C_i \\le... | 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 $x, y, z$ be complex numbers satisfying the following system of equations: \n\\[ \n\\begin{align*} \nz^2 + 5x &= 10z, \... | 40 |
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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 how many integers $n \\neq 1$ does $(n-1)^3$ divide $n^{2018(n-1)} - 1$?\n\nRemember to put your answer on its own line af... | 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\nHow many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60} = 1?$$\n\nRemember to put your answer on its ow... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $\\frac{1}{1}, \\frac{1}{2}, \\ldots, \\frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two number... | 2012 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA four-digit positive integer is called *virtual* if it has the form $\\overline{abab}$, where $a$ and $b$ are digits and $a \... | 8282 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered four-tuples of integers \\((a, b, c, d)\\) with \\(0 < a < b < c < d < 500\\) satisfy the conditions:\n1. \\(... | 870 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of positive integers $n \\leq 1000$ such that $\\text{lcm}(n, 9)$ is a perfect square. (Recall that $\\text... | 43 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,b$ be positive integers satisfying\n$$\\sqrt{\\frac{ab}{2b^2-a}}=\\frac{a+2b}{4b}$$.\nFind $|10(a-5)(b-15)|+8$.\n\nReme... | 2018 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many pairs of integers \\((a,b)\\) are there such that \\(0 \\leq a < b \\leq 100\\) and such that \\(\\frac{2^b-2^a}{2016... | 720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor an ordered $10$-tuple of nonnegative integers $a_1,a_2,\\ldots, a_{10}$, we denote:\n$$f(a_1,a_2,\\ldots,a_{10})=\\left(\\... | 462 |
[
{
"content": "Solve the following math problem step by 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 nonnegative integer $n$, the following are true:\n- $f(0) = 0$\n- $f(1) = 1$\n- $f(n) = f\\left(n - \\frac{m(m-1)}{2}\\rig... | 646 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe expression \\( \\frac{(1+2+\\cdots + 10)(1^3+2^3+\\cdots + 10^3)}{(1^2+2^2+\\cdots + 10^2)^2} \\) reduces to \\( \\frac{m}... | 104 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are real numbers $x$, $y$, and $z$ such that the value of \\(x + y + z - \\left(\\frac{x^2}{5} + \\frac{y^2}{6} + \\frac... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA **binary palindrome** is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same ba... | 92 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA function $f(x)$ satisfies $f(x) = f\\left(\\frac{c}{x}\\right)$ for some real number $c > 1$ and all positive numbers $x$. ... | 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\nFind the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p > 3$, we hav... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the expression:\n\\[\n \\frac{121 \\left( \\frac{1}{13} - \\frac{1}{17} \\right) \n + 169 \\left( \\frac{1}... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAs shown in the picture, the triangles $ABC$, $ADE$, and $EFG$ are all isosceles right triangles, such that $\\angle ACB = \\a... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a circle, 2022 points are chosen such that the distance between any two adjacent points is the same. There are $k$ arcs, ea... | 1011 |
[
{
"content": "Solve the following math problem step by 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 equation $x^3 - 9x^2 + 8x + 2 = 0$ has three real roots $p$, $q$, $r$. Find $\\frac{1}{p^2} + \\frac{1}{q^2} + \\frac{1}{r... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits?... | 199 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than $1998$ are relatively prime to $1547$? Two integers are relatively prime if they have no ... | 1487 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ways to completely fill a $3 \\times 15$ rectangle with non-overlapping $1 \\times 3$ rectangles.\n\nRem... | 189 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five pos... | 337 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number... | 1801 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that\n\\[\n\\prod_{n=1}^{\\infty}\\left(\\frac{1+i\\cot\\left(\\frac{n\\pi}{2n+1}\\right)}{1-i\\cot\\left(\\frac{n\\pi... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find \\[ \\frac{bc}{a^2} + \\frac{ac}{b^2} + \\frac{ab}{c^2}.... | -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 $Answer is the answer to the problem.\n\nFind the least prime factor of the number \\( \\frac{2016^{2016}-3}{3} \\).\n\nRemember to put your answer on its own line aft... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor primes $a$, $b$, and $c$ that satisfy the following conditions, calculate $abc$:\n\n1. $b + 8$ is a multiple of $a$.\n2. $... | 2009 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCDEF$ be a regular hexagon with side length $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQ... | 72 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, where $AB=2$, $AC=1+\\sqrt{5}$, and $\\angle CAB=54^{\\circ}$, suppose point $D$ lies on the extension of $... | 63 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \\in \\mathbb{R}$.\n\nRemember to... | 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\nFor a positive integer $k$, let $d(k)$ denote the number of divisors of $k$, and let $s(k)$ denote the digit sum of $k$. A pos... | 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 $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. Suppose $AD, BE, CF$ are concurrent at $P... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $D$, $E$, and $F$ be the feet of the altitudes from vertices $A$, $B$, and $C$ of an acute triangle $\\triangle ABC$, resp... | 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\nfantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants ... | 421 |
[
{
"content": "Solve the following math problem step by 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 have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among ... | 26 |
[
{
"content": "Solve the following math problem step by 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 $1, 2, \\ldots, 49, 50$ are written on the blackboard. Ann performs the following operation: she chooses three arb... | 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\nBalls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there wou... | 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 $Answer is the answer to the problem.\n\nJeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is gre... | 101 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTony's favorite \"sport\" is a spectator event known as the $\\textit{Super Mega Ultra Galactic Thumbwrestling Championship}$ ... | 1007012 |
[
{
"content": "Solve the following math problem step by 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 $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both p... | 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\nCompute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \\[(ax+by)^3 + (cx+dy)^3 = x^3 + y^3\\] hold... | 18 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, rectangle $ABCD$ has $AB = 5$ and $AD = 12$. Also, $E$ is a point in the same plane outside $ABCD$ such ... | 145 |
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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 us consider a polynomial $P(x)$ with integer coefficients satisfying:\n\\[\nP(-1) = -4, \\\\\nP(-3) = -40, \\\\\nP(-5) = -... | 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 $A$ be a set of numbers chosen from $1, 2, \\ldots, 2015$ with the property that any two distinct numbers, say $x$ and $y$... | 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\nSam Wang decides to evaluate an expression of the form $x + 2 \\cdot 2 + y$. However, he unfortunately reads each 'plus' as a ... | 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\nIn triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\\overline{BC}$, and let line $BE$ meet ... | 55 |
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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 picture below shows how Juan wants to divide a square field into three regions, all sharing a well at vertex $B$. If the s... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $X = (x_1, x_2, \\ldots, x_9)$ be a permutation of the set $\\{1, 2, \\ldots, 9\\}$. Let $A$ be the set of all such permut... | 121 |
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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 $Answer is the answer to the problem.\n\nOn Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most... | 1009 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\\{r\\}$ the fractional part of $r$. For ex... | 2837 |
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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 circles $\\omega_1$ and $\\omega_2$. They intersect in two points, one of which is the point $A$. Point $B$ lies on ... | 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 $Answer is the answer to the problem.\n\nA semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as ... | 540 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations:\n\\[\n\\begin{align*}\n(a+b)(c+... | 214 |
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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 $Answer is the answer to the problem.\n\nYoung Guy likes to make friends with numbers, so he calls a number \"friendly\" if the sum of its digits is equal to the produ... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n**Problem 3.** A three-digit number ending in $2$ is written on the board. If we erase the $2$ and place it at the beginning o... | 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\nGiven a positive integer $N$ (written in base $10$), define its integer substrings to be integers that are equal to strings of... | 88888888 |
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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 has exactly $81$ divisors, which are arranged in a $9 \\times 9$ table. In this table, for any two numbers ... | 2 |
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