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\nCarrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair... | 120 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a regular hexahedron and a regular octahedron, all their faces are regular triangles with each side of length $a$. Let the... | 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\nThe remainder when $x^{100} - x^{99} + \\ldots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a ... | -49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle has sides of length $\\sqrt{13}$, $\\sqrt{17}$, and $2 \\sqrt{5}$. Compute the area of the triangle.\n\nRemember to... | 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 $Answer is the answer to the problem.\n\nKnowing that the system:\n\\[\n\\begin{align*}\nx + y + z &= 3, \\\\\nx^3 + y^3 + z^3 &= 15, \\\\\nx^4 + y^4 + z^4 &= 35,\n\\e... | 83 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle $\\triangle A_0A_1A_2$ in the plane has side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, $A_2A_0 = 9$. For $i \\ge 0$, give... | 422 |
[
{
"content": "Solve the following math problem step by 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 distinct possible values of $x^2 - 4x + 100$, where $x$ is an integer between 1 and 100, inclusive.\n\nRem... | 328053 |
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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 $a_1 = 24$ and form the sequence $a_n$, for $n \\geq 2$, by the recurrence relation $a_n = 100a_{n-1} + 134$. The first fe... | 88 |
[
{
"content": "Solve the following math problem step by 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 quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. During the first week,... | 3000 |
[
{
"content": "Solve the following math problem step by 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 $AB = 14$, $BC = 30$, $AC = 40$ and $\\triangle AB'C'$ with $AB' = 7\\sqrt{6}$, $B'C' = 15\\sqrt{6}$... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest positive integer that is a divisor of all three of \\( A = 2^{2010} \\times 3^{2010} \\), \\( B = 3^{201... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a square with side length $16$ and center $O$. Let $\\mathcal{S}$ be the semicircle with diameter $AB$ that lies... | 136 |
[
{
"content": "Solve the following math problem step by 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 $\\mathbb{Z}^{\\geq 0}$ be the set of all non-negative integers. Consider a function $f: \\mathbb{Z}^{\\geq 0} \\to \\math... | 2045254 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of sequences of positive integers $1 = x_0 < x_1 < \\dots < x_{10} = 10^{5}$ with the property that for e... | 252 |
[
{
"content": "Solve the following math problem step by 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 $a_{1} < a_{2}< \\cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \\cdots <b_{20... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \\leq i \\leq 4$. We ar... | 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\nIf $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\\triangle CDX$ is equilateral, compute $\\angl... | 84 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, and $BD = 24$. If $r$ is the radius of the circle ... | 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 $Answer is the answer to the problem.\n\nNaomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of t... | 540 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of even integers \\( n \\) such that \\( 0 \\le n \\le 100 \\) and \\( 5 \\mid n^2 \\cdot 2^{2n^2} + 1 \\).\n\... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the last four decimal digits of the number $1997 \\cdot 5^{1998}$.\n\nRemember to put your answer on its own line af... | 3125 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the highest power of $2$ that divides exactly into $1996! = 1 \\times 2 \\times \\cdots \\times 1996$.\n\nRemember to put... | 1989 |
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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 contest consists of six problems, each worth 7 points. For any given problem, a contestant can score either $0$, $1$, or $7$... | 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\nLet $a_1 < a_2 < \\cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime t... | 1054 |
[
{
"content": "Solve the following math problem step by 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 natural number multiples of $7$, and each side is smaller than $40$. How many triangles sat... | 22 |
[
{
"content": "Solve the following math problem step by 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 is considered **happy** if:\n\n1. All its digits are different and not $0$.\n2. One of its digits is equal ... | 264 |
[
{
"content": "Solve the following math problem step by 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 $20^{21} = 2^a 5^b = 4^c 5^d = 8^e 5^f$ for positive integers $a, b, c, d, e,$ and $f$. Find $\\frac{100bdf}{ace}... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a quadratic function that satisfies the following condition. Find the value of $\\frac{f(8)-f(2)}{f(2)-f(1)}$.\n\nF... | 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\nThe four sets $A$, $B$, $C$, and $D$ each have $400$ elements. The intersection of any two of the sets has $115$ elements. The... | 1094 |
[
{
"content": "Solve the following math problem step by 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 natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \\leq n$, define an $n,s,k$-block as a st... | 128 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the maximum possible value of $9\\sqrt{x} + 8\\sqrt{y} + 5\\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers sati... | 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 the sum of all the positive integers that are divisors of either $96$ or $180$.\n\nRemember to put your answer on its own... | 770 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both ... | 25150 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of digits in $2015!$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user... | 5786 |
[
{
"content": "Solve the following math problem step by 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 $n < 2549$ are there such that $x^2 + x - n$ has an integer root?\n\nRemember to put your answer on... | 49 |
[
{
"content": "Solve the following math problem step by 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 positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \\[a^2 + b^2... | 594 |
[
{
"content": "Solve the following math problem step by 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 sequences of integers $(a_1, \\ldots, a_{200})$ such that the following conditions hold:\n\n1. $0 \\leq ... | 20503 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe repeating decimal $2.0151515\\ldots$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive i... | 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\nCompute the number of integer solutions $(x, y)$ to the equation $xy - 18x - 35y = 1890$.\n\nRemember to put your answer on it... | 96 |
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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 intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance ... | 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\nWhat is the largest positive integer $n$ such that $10 \\times 11 \\times 12 \\times \\ldots \\times 50$ is divisible by $10^n... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe call $\\triangle ABC$ with centroid $G$ \"balanced\" on side $AB$ if the foot of the altitude from $G$ onto line $\\overlin... | 4046 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \\overline{aa}$, which is a ... | 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\nFind the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\\sqrt[3]{n}$.\n\nRem... | 420 |
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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 $a$ and $b$ be integer solutions to $17a + 6b = 13$. What is the smallest possible positive value for $a - b$?\n\nRemember... | 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\nYou have a $10 \\times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$, $\\ldots$, ... | 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 $Answer is the answer to the problem.\n\nA village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is locate... | 48 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA store offers packages of $12$ pens for $\\$10$ and packages of $20$ pens for $\\$15$. Using only these two types of packages... | 224 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a... | -2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three di... | 90720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divi... | 880 |
[
{
"content": "Solve the following math problem step by 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 $\\triangle ABC$, $\\angle A = 30^{\\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\\frac{DB}{DC... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f:\\{1,2,\\dots,2019\\}\\to\\{-1,1\\}$ be a function, such that for every $k\\in\\{1,2,\\dots,2019\\}$, there exists an $... | 673 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive persons of different heights stand next to each other on numbered booths to take a picture. In how many ways can they be ... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ and $DEF$ be two triangles such that $AB = DE = 20$, $BC = EF = 13$, and $\\angle A = \\angle D$. If $AC - DF = 10$,... | 126 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $20^{20} + 21^{21} - 21^{20} - 20^{21}$ is divided by 100.\n\nRemember to put your answer on its own l... | 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\nFor any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n ... | 632 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $a, b, c$ are the roots of the polynomial $x^3 + 20x^2 + 1x + 5$, compute $(a^2 + 1)(b^2 + 1)(c^2 + 1)$.\n\nRemember to put... | 225 |
[
{
"content": "Solve the following math problem step by 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 $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible va... | -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\nWe color each number in the set $S = \\{1, 2, \\ldots, 61\\}$ with one of $25$ given colors. It is not necessary for every col... | 119 |
[
{
"content": "Solve the following math problem step by 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 equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. The bands are co... | 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\nKatherine makes Benj play a game called \\(50\\) Cent. Benj starts with \\(\\$0.50\\), and every century thereafter has a \\(5... | 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\nThe roots of the equation $x^3 - x + 1 = 0$ are $a$, $b$, and $c$. Find $a^8 + b^8 + c^8$.\n\nRemember to put your answer on i... | 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 straight line $y = ax + 16$ intersects the graph of $y = x^3$ at 2 distinct points. What is the value of $a$?\n\nRemember ... | 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\nThe real numbers $x$ and $y$ satisfy the following system of equations:\n\\[\n\\begin{cases} \n\\sqrt{3x}\\left(1+\\dfrac{1}{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 $Answer is the answer to the problem.\n\nConsider all ordered pairs \\((a, b)\\) of positive integers such that \\(\\frac{a^2 + b^2 + 2}{ab}\\) is an integer and \\(a ... | 52 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\\frac... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65 \\).\nLet the roots of \\( f(x) \\) be \\( a_k + ib_k \\) for \\( k = 1, 2, 3, 4 ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many real numbers are roots of the polynomial \\[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\\]?\n\nRemember to ... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers... | 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 $ABCD$ be an isosceles trapezoid with $AD \\parallel BC$. Points $P$ and $Q$ are placed on segments $\\overline{CD}$ and $... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1$, $d(i,1) = 1$ for all $i$ and $j$, and for $i, j >... | 481 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be the number of distinct rearrangements of the 34 letters in \"SUPERCALIFRAGILISTICEXPIALIDOCIOUS\". How many positiv... | 3225600 |
[
{
"content": "Solve the following math problem step by 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 starts with the number $0$. She can apply $100$ operations on her number. In each operation, she can either add $1$ to h... | 94 |
[
{
"content": "Solve the following math problem step by 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 sum of the decimal digits of a natural number $n$ equals 100, and the sum of the digits of $44n$ equals 800. Determine the... | 300 |
[
{
"content": "Solve the following math problem step by 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 define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$. What is\n\\[ \\left\\lfloor \\frac{5^{201701... | 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 $Answer is the answer to the problem.\n\nAlice has three daughters, each of whom has two daughters. Each of Alice's six grand-daughters has one daughter. How many sets... | 2926 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvin's calculator is broken and can only perform three operations:\n\n1. Given a number $x$, output $2x$.\n2. Given a number $... | 82 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider a sequence $x_1, x_2, \\cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \\dots, 10$, let \\[ x_... | 622 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.\n\nRemember to put your answer on its own line ... | 504 |
[
{
"content": "Solve the following math problem step by 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 $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Let $\\Gamma$ denote the circumcircle of $\\triangle ABC$. Let ... | 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\nFor a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Fi... | 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\nMatt has somewhere between $1000$ and $2000$ pieces of paper that he is trying to divide into piles of the same size. He attem... | 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\nHow many functions $f : \\{1,2,3,4,5,6\\} \\to \\{1,2,3,4,5,6\\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by ... | 64 |
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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 $Answer is the answer to the problem.\n\nThe figure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles.... | 252 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the t... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the least possible value of $f(1998)$, where $f: \\mathbb{N} \\to \\mathbb{N}$ is a function such that for all $m, n... | 120 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ is called bad if it cannot be expressed as the product of two distinct positive integers greater than $... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airp... | 512 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the first digit after the decimal point of the number $\\displaystyle \\frac{1}{1009} + \\frac{1}{1010} + \\cdots + \\fra... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its ma... | 85 |
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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 circles with radii $1$ and $2$ that are externally tangent to each other and internally tangent to a c... | 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\nWhat is the largest prime factor of $4^9 + 9^4$?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role... | 881 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the tr... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n \\geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \\[1 + z + z... | 404 |
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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 $x_0, x_1, \\ldots$ be a sequence of real numbers such that $x_n = \\frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \\geq 2$. Find ... | 128 |
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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 $f(x)=\\int_0^{x} e^{t} (\\cos t+\\sin t)\\, dt$, $g(x)=\\int_0^{x} e^{t} (\\cos t-\\sin t)\\, dt$. \n\nFor a real number ... | 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\nTwo toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5... | 19152 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a plane, there is an invisible rabbit hiding on a lattice point. We want to place $n$ hunters on some lattice points to cat... | 4 |
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