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 $Answer is the answer to the problem.\n\nWe flip a fair coin 10 times. What is the probability that we get heads in at least 6 of the 10 flips?The answer is in the fo... | 705 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ having the property that for any $n$ distinct integers $a_1, a_2, \\dots , a_n$ the pro... | 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 $Answer is the answer to the problem.\n\nCompute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.\n\nRe... | 25617208995 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections ... | 36 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.... | 324 |
[
{
"content": "Solve the following math problem step by 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 positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decima... | 32768 |
[
{
"content": "Solve the following math problem step by 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 $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.\n\nRemember to pu... | 2000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not gr... | 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 $Answer is the answer to the problem.\n\nFor an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics pr... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider a set $ A$ of positive integers such that the least element of $ A$ equals $ 1001$ and the product of all elements of... | 1040 |
[
{
"content": "Solve the following math problem step by 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 $k$ and $n$ be positive integers and let $x_1, x_2, \\cdots, x_k, y_1, y_2, \\cdots, y_n$ be distinct integers. A polynomi... | 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\nA mason has bricks with dimensions $2\\times5\\times8$ and other bricks with dimensions $2\\times3\\times7$. She also has a bo... | 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 $Answer is the answer to the problem.\n\n$ S$ is a non-empty subset of the set $ \\{ 1, 2, \\cdots, 108 \\}$, satisfying:\n\n(1) For any two numbers $ a,b \\in S$ ( ma... | 76 |
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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 $m$ which makes it possible to color several cells of a $70\\times 70$ table red such that [... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob’s Rice ID number has six digits, each a number from $1$ to $9$, and any digit can be used any number of times. The ID numb... | 324 |
[
{
"content": "Solve the following math problem step by 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 even positive integers less than $233$ not divisible by $10$.\n\nRemember to put your answer on its own li... | 10812 |
[
{
"content": "Solve the following math problem step by 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 |
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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 man named Juan has three rectangular solids, each having volume $128$. Two of the faces of one solid have areas $4$ and $32$... | 688 |
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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\nLet $f(x)=x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cdots}}}$. Find $f(99)f^\\prime (99)$.\n\nRemember to put your answer... | 99 |
[
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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 $Answer is the answer to the problem.\n\nYou are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open a... | 9 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $C_1$ and $C_2$ be two concentric reflective hollow metal spheres of radius $R$ and $R\\sqrt3$ respectively. From a point ... | 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 $Answer is the answer to the problem.\n\nJohny's father tells him: \"I am twice as old as you will be seven years from the time I was thrice as old as you were\". What... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCalculate the number of ways of choosing $4$ numbers from the set ${1,2,\\cdots ,11}$ such that at least $2$ of the numbers ar... | 274 |
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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\nEvaluate $2000^3-1999\\cdot 2000^2-1999^2\\cdot 2000+1999^3$\n\nRemember to put your answer on its own line after \"Answer:\".... | 3999 |
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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\nFive students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible dif... | 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\nDetermine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.\n\nRemember to put your answer on its own line after \"... | 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\nAll arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, w... | 738826 |
[
{
"content": "Solve the following math problem step by 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 $\\cos (\\alpha + \\beta) + sin (\\alpha - \\beta) = 0$, $\\tan \\beta =\\frac{1}{2000}$, find $\\tan \\alpha$.\n\nRemem... | -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 $[ \\sqrt{19992000}]$ where $[a]$ is the greatest integer less than or equal to $x$.\n\nRemember to put your answer on it... | 4471 |
[
{
"content": "Solve the following math problem step by 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 $a$ such that $x^4+a^2$ is not prime for any integer $x$.\n\nRemember to put your answer on... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all polynomials $f(x)$ with integer coefficients such that the coefficients of both $f(x)$ and $[f(x)]^3$ lie in the set ... | 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\nHow many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities?\n$$a^2 + b^2 < 16$$\n$$a^2 + b^2 < 8a$$... | 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 $Answer is the answer to the problem.\n\n$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily con... | 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\nRight triangle $XY Z$ has right angle at $Y$ and $XY = 228$, $Y Z = 2004$. Angle $Y$ is trisected, and the angle trisectors in... | 1370736 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA classroom consists of a $5\\times5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student wi... | 962 |
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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 $f(x)=\\sin(\\sin(x))$. Evaluate \\[ \\lim_{h \\to 0} \\dfrac {f(x+h)-f(h)}{x} \\] at $x=\\pi$.\n\nRemember to put your an... | 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 the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.\n\nRemember to put your answer on its o... | 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\nThe number $27,\\,000,\\,001$ has exactly four prime factors. Find their sum.\n\nRemember to put your answer on its own line a... | 652 |
[
{
"content": "Solve the following math problem step by 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 dot is marked at each vertex of a triangle $ABC$. Then, $2$, $3$, and $7$ more dots are marked on the sides $AB$, $BC$, and ... | 357 |
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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 $Answer is the answer to the problem.\n\nVernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, ... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also ... | 33 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\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 $AP=6$, and... | 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 $Answer is the answer to the problem.\n\nEight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list ... | 3507 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe are given some similar triangles. Their areas are $1^2,3^2,5^2,\\cdots,$ and $49^2$. If the smallest triangle has a perim... | 2500 |
[
{
"content": "Solve the following math problem step by 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 $A_{10}$ denote the answer to problem $10$. Two circles lie in the plane; denote the lengths of the internal and external ... | 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\nCircle $\\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tangents... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x,y,n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^n-... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA student at Harvard named Kevin\nWas counting his stones by $11$\nHe messed up $n$ times\nAnd instead counted $9$s\nAnd wound... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence consists of the digits $122333444455555\\ldots$ such that the each positive integer $n$ is repeated $n$ times, in i... | 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\nDetermine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^n$.\n\nRemember to put your answer on its own line... | 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\nIn triangle $ABC$, $\\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\\angle ABD$ is right, and point $E$ lies ... | 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 number of $7$-tuples $(n_1,\\ldots,n_7)$ of integers such that \\[\\sum_{i=1}^7 n_i^6=96957.\\]\n\nRemember to put yo... | 2688 |
[
{
"content": "Solve the following math problem step by 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 circumradius of cyclic hexagon $ABCDEF$, which has side lengths $AB=BC=2$, $CD=DE=9$, and $EF=FA=12$.\n\nRemember ... | 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 $ ABC$ be a triangle with $ BC \\equal{} 2007$, $ CA \\equal{} 2008$, $ AB \\equal{} 2009$. Let $ \\omega$ be an excircle ... | 2014024 |
[
{
"content": "Solve the following math problem step by 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 mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (bot... | 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\nDetermine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\\le a_k\\le k$... | 1540 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $ x^3 \\minus{} 9x^2 \\plus{} 8x \\plus{} 2 \\equal{} 0$ has three real roots $ p$, $ q$, $ r$. Find $ \\frac {1... | 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\nDetermine the number of ways to select a sequence of $ 8$ sets $ A_1,A_2,\\ldots,A_8$, such that each is a subset (possibly em... | 2025 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many ordered triples $ (a,b,c)$ of positive integers are the equations $ abc\\plus{}9 \\equal{} ab\\plus{}bc\\plus{}ca... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of subsets $ S$ of $ \\{1,2, \\dots 63\\}$ the sum of whose elements is $ 2008$.\n\nRemember to put your answe... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe squares of a $3\\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, ... | 2448 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) =x^4+14x^3+52x^2+56x+16.$ Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $5$ students on a team for a math competition. The math competition has $5$ subject tests. Each student on the team ... | 2040 |
[
{
"content": "Solve the following math problem step by 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 $a,b,c,x,y,$ and $z$ be complex numbers such that \\[a=\\dfrac{b+c}{x-2},\\qquad b=\\dfrac{c+a}{y-2},\\qquad c=\\dfrac{a+b... | -5892 |
[
{
"content": "Solve the following math problem step by 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 sphere is the set of points at a fixed positive distance $r$ from its center. Let $\\mathcal{S}$ be a set of $2010$-dimensio... | 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\nSuppose that a polynomial of the form $p(x)=x^{2010}\\pm x^{2009}\\pm \\cdots \\pm x \\pm 1$ has no real roots. What is the m... | 1005 |
[
{
"content": "Solve the following math problem step by 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 polynomials of degree exactly $5$ with real coefficients send the set $\\{1, 2, 3, 4, 5, 6\\}$ to a permutation of it... | 714 |
[
{
"content": "Solve the following math problem step by 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 values of... | -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\nLet $A = \\{1,2,\\ldots,2011\\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \\le n$ for all $n$ in ... | 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\nSuppose that there are $16$ variables $\\{a_{i,j}\\}_{0\\leq i,j\\leq 3}$, each of which may be $0$ or $1$. For how many sett... | 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\nHow many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots,... | 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\nDefine the sequence of positive integers $\\{a_n\\}$ as follows. Let $a_1=1$, $a_2=3$, and for each $n>2$, let $a_n$ be the r... | 23097 |
[
{
"content": "Solve the following math problem step by 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 real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equ... | -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 the number of positive divisors $d$ of $15!=15\\cdot 14\\cdot\\cdots\\cdot 2\\cdot 1$ such that $\\gcd(d,60)=5$.\n\nRemem... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many unordered sets $\\{a,b,c,d\\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z... | 761474 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups neve... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial ... | -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\nFind the rightmost non-zero digit of the expansion of $(20)(13!)$.\n\nRemember to put your answer on its own line after \"Answ... | 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\nCompute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\\leq a_1,a_2,a_3,a_4,a_5... | 6528 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNatalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interva... | 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\nEli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following c... | 315 |
[
{
"content": "Solve the following math problem step by 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 calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\\lfloor \\fra... | 233 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1,a_2,\\ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i\\geq 1$, and let $b_i$ ... | 127 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value of $a+b+c... | 153 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB=2$, $AC=1+\\sqrt{5}$, and $\\angle CAB=54^{\\circ}$. Suppose $D$ lies on the extension of $AC$ through ... | 63 |
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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\t For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by \n\\[\nW(n,k) = \n\\begin{case... | 875 |
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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 infinite sequence of real numbers $a_1, a_2, \\dots$ satisfies the recurrence\n\\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \\]\n... | 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\n\n\t For a positive integer $n$, denote by $\\tau(n)$ the number of positive integer divisors of $n$, and denote by $\\phi(n)$... | 172 |
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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\t Let $a,b$ and $c$ be real numbers such that\n\\begin{align*}\na^2+ab+b^2&=9 \\\\\nb^2+bc+c^2&=52 \\\\\nc^2+ca+a^2&=49.\n\... | 52 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\n\t A contest has six problems worth seven points each. On 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\n\n\t How many equilateral hexagons of side length $\\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at latti... | 216 |
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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 positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a p... | 329 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the remainder when\n$$\\sum_{i=0}^{2015} \\left\\lfloor \\frac{2^i}{25} \\right\\rfloor$$\nis divided by 100, where ... | 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\n\n\t On the Cartesian plane $\\mathbb{R}^2$, a circle is said to be \\textit{nice}\nif its center is at the origin $(0,0)$ and... | 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\t Define the sequence $a_1, a_2 \\dots$ as follows: $a_1=1$ and for every $n\\ge 2$,\n\\[\na_n = \\begin{cases}\nn-2 & \\text... | 51 |
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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 $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Fin... | 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\nLet $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides... | 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\nThere are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one c... | 2014 |
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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\nAt a recent math contest, Evan was asked to find $2^{2016} \\pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan ... | 211 |
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