prompt listlengths 1 1 | label stringlengths 1 210 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe grid below contains five rows with six points in each row. Points that are adjacent either horizontall... | 14 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of $8$-tuples $(\\epsilon_1, \\epsilon_2, \\ldots, \\epsilon_8)$ such that $\\epsilon... | 88 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the regular hexagon $ABCDEF$, a point $X$ is marked on the diagonal $AD$ such that $\\angle AEX = 65^\\... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ mi... | 89 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and... | 292 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, \"If I give you some ... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n \\ge 2$, define $k(n)$ to be the largest integer $m$ such that $(n!)^m$ divid... | 89 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. ... | 217 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases w... | 28812 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest positive integer $n$ such that the following equations have integer solutions in $x, y_1,... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all non-negative integers $k$ such that there exists a function $f : \\mathbb{N} \\to \\mathbb{N... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers $x$, where $x \\ge 3$, such that $201020112012_x$ (interpreted as a base $x$ ... | 32 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 triples of positive integers $(a, b, c)$ with $1 \\leq a, b, c \\leq 5$ is the quantity $(a+b... | 48 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \\(5^m + 63n + 49 = a^... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 function $f(x) = ax^2 + bx + c$ satisfies the following conditions:\n1. $f(\\sqrt{2}) = 3$\n2. $|f(x)|... | 4025 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe integers $1, 2, \\ldots, n$ are written in order on a long slip of paper. The slip is then cut into fi... | 2014 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 of integral points on the circle with center $ (199, 0) $ and radius $ 199 $ is ________.\n\nRe... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $x, y,$ and $z$ be nonnegative real numbers with $x+y+z=120$. Compute the largest possible value of th... | 160 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $(a, b)$ of positive integers with $1 \\le a, b \\le 10$ are there such that in the... | 33 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $\\triangle AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on ... | 5300 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNumber $125$ is written as the sum of several pairwise distinct and relatively prime numbers, greater than... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nInitially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar chooses a divis... | 2033 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $k$ is $2014$-ambiguous if the quadratics $x^2 + kx + 2014$ and $x^2 + kx - 2014$ both ... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive... | 4489 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 remainder when $10^{2021}$ is divided by $10101$.\n\nRemember to put your answer on its own li... | 9091 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ways to erase 24 letters from the string ``OMOMO\\cdots OMO'' (with length 27), such... | 455 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $x$ and $y$ are positive real numbers such that $\\frac{5}{x} = \\frac{y}{13} = \\frac{x}{y}$, ... | 1170 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n16 chess players held a tournament among themselves where every two players played exactly one game. The s... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, and $c$ be real numbers such that $0 \\le a, b, c \\le 5$ and $2a + b + c = 10$. Over all po... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 4-digit numbers $\\overline{abcd}$ that satisfy the following conditions:\n\n1. The number is a m... | 3454 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nShenelle has some square tiles. Some of the tiles have side length $5\\text{ cm}$ while the others have si... | 94 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \\[ \\frac{1}{a} + \\frac{1}{b} + \\f... | 56 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$,... | 90 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1) = 91$ and $P(-1) = -121$, compu... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and need... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nComplex number $\\omega$ satisfies $\\omega^5 = 2$. Find the sum of all possible values of $\\omega^4 + \\... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the remainder when $2^{1023}$ is divided by $1023$?\n\nRemember to put your answer on its own line... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any ... | 324 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 grid of points $X = \\{(m,n) \\mid 0 \\leq m,n \\leq 4 \\}$. We say a pair of points $\\{(a,b... | 48 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator o... | 168 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAna has \\(22\\) coins. She can take from her friends either \\(6\\) coins or \\(18\\) coins, or she can g... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the p... | 4825 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered triplets $(a, b, c)$ of positive integers such that $abc = 2008$.\n\nRemember t... | 30 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $x$, $y$, and $z$ are integers that satisfy the system of equations: \n\\[ x^2y + y^2z + z^2x = 2... | 245 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 regular $2n + 1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n... | 8000 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many ways can Irena arrange the integers $1$ through $7$ in a line such that for any three consecut... | 232 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $a_n$ be the integer closest to $\\sqrt{n}$. Find the sum $$\\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots ... | 88 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 of the $9$ small circles of the following figure, write positive integers less than $10$, without ... | 28 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $x^{x^4} = 4$, what is the value of $x^{x^2} + x^{x^8}$?\n\nRemember to put your answer on its own line... | 258 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:\n\\[\n... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 five-letter \"words\" can you spell using the letters $S$, $I$, and $T$, if a \"word\" is defined... | 123 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTalithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month ... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $\\angle A = 45^\\circ$ and $M$ is the midpoint of $\\overline{BC}$. The line $\\overli... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPatrick started walking at a constant rate along a straight road from school to the park. One hour after P... | 277 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMoor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of wh... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose... | 31 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind a number with $3$ digits, knowing that:\n1. The sum of its digits is $9$.\n2. The product of its digi... | 342 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 trinomial $f(x)$ satisfies the equation $(f(x))^3 - f(x) = 0$, which has three real roots. Determine t... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRichard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday... | 1100 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSet \n\\[A_n=\\sum_{k=1}^n \\frac{k^6}{2^k}.\\]\nFind \\(\\lim_{n\\to\\infty} A_n.\\)\n\nRemember to put y... | 9366 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?\n\... | 63 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none o... | 1225 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 real numbers that satisfy the following equations:\n\n\\[\n(x-y)^2 + (y-z)^2 + (z-x)^... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ which can be expressed in the form $k = 19^n - 5^m$ for some positi... | 14 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoint $P$ is in the interior of $\\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = ... | 600 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe **doubling sum** function is defined by:\n\\[ D(a,n) = a + 2a + 4a + 8a + \\ldots \\] \nThis sum conta... | 9765 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers... | 2713 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ such that $k!$ ends in at least 43 zeroes.\n\nRemember to put your ... | 175 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between ... | 98 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWill has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times ... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 integers $x$ satisfy the following conditions?\n\n1. $|x| + 5 < 7$\n2. $|x - 3| > 2$\n\nRemember ... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou come across an ancient mathematical manuscript. It reads: \n\n\"To determine if a number is divisible ... | 106 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the $2019$th strictly positive integer $n$ such that $\\binom{2n}{n}$ is not divisible by $5$.\n\nRem... | 37805 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 circles have radii $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the interse... | 24 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 positive integers expressible in the form $\\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of triangles of different sizes which contain the gray triangle in the figure below.\n\... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of all unordered pairs $\\{A,B \\}$ of subsets of an $8$-element set, such that $A \\cap B... | 23606 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any positive integer $a$, let $\\tau(a)$ be the number of positive divisors of $a$. Find, with proof, ... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJonathan discovers that his ideal match is Sara Lark. To increase his chances of finding a girlfriend, he ... | 23520 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the value of the sum \\( \\left| \\sum_{1 \\leq i < j \\leq 50} ij(-1)^{i+j} \\right|.\\)\n\nRem... | 21150 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn Vila Par, all the truth coins weigh an even number of grams, while the false coins weigh an odd number ... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.... | 104 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGorf the frog is standing on the first lily pad in a row of lily pads numbered from $1$ to $20$ from left ... | 25 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest natural number $n$ for which there exists a natural number $x$ such that\n\\[(x+1)^3 + (... | 10 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 function such that $f(x + y) = f(x) + f(y)$ for all $x, y \\in \\mathbb{R}$ and $f(1) = 100$.... | 403791300 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the leftmost three digits of the number \\[1^1 + 2^2 + 3^3 + \\ldots + 999^{999} + 1000^{1000}.\... | 100 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficient... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 positive integers $v, w, x, y$, and $z$ satisfy the equation \\[v + \\frac{1}{w + \\frac{1}{x + \\frac... | 12354 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$... | 48 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 is a rectangular field that measures $20\\,\\text{m}$ by $15\\,\\text{m}$. Xiaoyu the butterfly is s... | 17 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of i... | 648 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x, y, z$ be positive real numbers that satisfy at least one of the inequalities: $2xy > 1$, $yz > 1$.... | -8 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $a_n = \\sum_{d|n} \\frac{1}{2^{d+ \\frac{n}{d}}}$. In other words, $a_n$ is the sum of $\\frac{1}{2^{... | 4 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven 10 real numbers $a_1, a_2, \\ldots, a_{10}$, 45 sums of two of these numbers are formed as $a_i + a_... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $12 \\times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two roo... | 73 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Stromquist Comet is visible every 61 years. If the comet was visible in 2017, what is the next leap ye... | 2444 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = (x-1)(x-2)(x-3)^2(x-4)(x-5)(x-6)$. Compute $f''(3) - f'(3) + f(3)$.\n\nRemember to put your an... | -24 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 side lengths $13$, $14$, and $15$. The points on the interior of $\\triangle ... | 193 |
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