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 $Answer is the answer to the problem.\n\nThe sum \\(\\sum_{k=3}^{\\infty} \\frac{1}{k(k^4-5k^2+4)^2}\\) is equal to \\(\\frac{m^2}{2n^2}\\), where ... | 49 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCarl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving webs... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1$, $a_2$, $\\cdots$ be a sequence such that $a_1 = a_2 = \\frac{1}{5}$, and for $n \\ge 3$, \n\\[... | 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\nTwo people $A$ and $B$ start from the same place at the same time to travel around a circular track of len... | 41 |
[
{
"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 a rectangular $57 \\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest p... | 457 |
[
{
"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 $Answer is the answer to the problem.\n\nJonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for... | 839 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDaeun draws a unit circle centered at the origin and inscribes within it a regular hexagon $ABCDEF$. Then ... | 101 |
[
{
"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 right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\\text{lcm}(a,b)=72$, $\\text{lc... | 3024 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, let $f(n)$ be the integer formed by reversing the digits of $n$ (and removing ... | 13 |
[
{
"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\nRegular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find ... | 84 |
[
{
"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 $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \\le n$, define \n\\[ S_m = \\sum_{\\... | 8027 |
[
{
"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 smallest subset of the integers with the property that $0 \\in S$ and for any $x \\in S$, w... | 128 |
[
{
"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 smallest integer $A$ such that there exist consecutive integers, including $A$, which sum to... | -2013 |
[
{
"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 $\\overline{ABCD}$ be a 4-digit number. What is the smallest possible positive value of $\\overline{AB... | 90 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ be the first term in the sequence $31, 331, 3331, \\ldots$ which is divisible by $17$. How many di... | 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\nSuppose $w, x, y, z$ satisfy the following equations:\n\\[\n\\begin{align*}\nw + x + y + z &= 25, \\\\\nwx... | 27 |
[
{
"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 $r_k$ denote the remainder when $\\binom{127}{k}$ is divided by $8$. Compute $r_1 + 2r_2 + 3r_3 + \\cd... | 2016 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\\lce... | 31 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1... | 1033 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a point on the graph of the function $y = x + \\frac{2}{x}$ for $x > 0$. The lines $PA$ and $PB... | -1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^3 - 7x^2 + 16x - 10$. As $x$ ranges over all integers, find the sum of distinct prime values... | 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\nThere are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20\\%$ of cat-own... | 126 |
[
{
"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 a cube, on each edge, write a number, either $1$ or $-1$. For each face of the cube, multiply the fo... | -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\nThe following sequence lists all the positive rational numbers that do not exceed $\\frac{1}{2}$ by first ... | 61 |
[
{
"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 $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given... | 660 |
[
{
"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 $M = \\{1,2,\\cdots , 10\\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different ... | 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\nThe sequence $(a_n)$ is defined by the following initial conditions and recurrence relations:\n\n- Initial... | 91 |
[
{
"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 value of $x$ where the graph of \\( y = \\log_3(\\sqrt{x^2+729}+x) - 2\\log_3(\\sqrt{x^2+729}-x) ... | 36 |
[
{
"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 $T$ be the set of all positive integer divisors of $2004^{100}$. What is the largest possible number o... | 10201 |
[
{
"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\nSets $A$, $B$, and $C$ satisfy the following conditions: \n- $|A| = 92$ \n- $|B| = 35$ \n- $|C| = 63$ ... | 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 $a, b, c$ be positive integers such that $a + 2b + 3c = 100$. Find the greatest value of $M = abc$.\n\... | 6171 |
[
{
"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\nOlya wrote fractions of the form $\\frac{1}{n}$ on cards, where $n$ represents all possible divisors of th... | 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's consider all possible natural seven-digit numbers, in the decimal notation of which the digits $1$, ... | 3651427 |
[
{
"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 positive integer $n$ such that:\n\\[\n\\underbrace{f(f(\\cdots f}_{2013 \\ f\\text{'s}}(n)\\cdots... | 1015057 |
[
{
"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 $Answer is the answer to the problem.\n\nA triangulation of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the... | 9089 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n > 1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n) = n... | 19 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor positive integers $i$ and $j$, define $d(i,j)$ as follows: \n- $d(1,j) = 1$ for all $j$ \n- $d(i,1) ... | 481 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAltitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $... | 33725 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be a polynomial with degree less than $4$ such that $p(x)$ attains a maximum at $x = 1$. If $p(1) ... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$... | 203 |
[
{
"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 $Answer is the answer to the problem.\n\nThree spheres are all externally tangent to a plane and to each other. Suppose that the radii of these sph... | 47 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \\frac{2\\Delta}{x} = y + \\frac{2\\... | 87 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine the function $f: \\mathbb{R} \\setminus \\{-1,1\\} \\to \\mathbb{R}$ as follows:\n\\[ f(x) = \\sum_... | 4033 |
[
{
"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 $a$ and $b$ are positive integers such that $3\\sqrt{2+\\sqrt{2+\\sqrt{3}}}=a\\cos\\frac{\\pi}{b}$, fin... | 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 $Answer is the answer to the problem.\n\n21 Savage has a 12-car garage, with a row of spaces numbered 1, 2, 3, \\ldots, 12. How many ways can he ch... | 357 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S = \\{1, 2, \\ldots, 8\\}$. How many ways are there to select two disjoint subsets of $S$?\n\nRememb... | 6561 |
[
{
"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 a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where each coefficient $a, b, c, d, e,$ and ... | 142 |
[
{
"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 $ABC$, given $AB = 2$, $AC = 1 + \\sqrt{5}$, and $\\angle CAB = 54^{\\circ}$. Suppose $D$ lies... | 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 $Answer is the answer to the problem.\n\nIn the diagram below, \\( \\overline{AB} \\) and \\( \\overline{CD} \\) are parallel, \\( \\angle BXY = 45... | 55 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^{2} + ax + b$. If for all nonzero real $x$:\n\\[ f\\left(x + \\frac{1}{x}\\right) = f\\left(... | 13 |
[
{
"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 a triangle $ABC$ with angles $\\angle A = 60^{\\circ}$, $\\angle B = 75^{\\circ}$, $\\angle C = 45^{... | 1132 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circum... | 43 |
[
{
"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 have two blackboards, $A$ and $B$. You need to write some integers on them, where each integer is grea... | 65 |
[
{
"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\nWe know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive int... | 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\nThe diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square... | 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 $Answer is the answer to the problem.\n\nCalculate \\(1^2 - 2^2 + 3^2 - 4^2 + \\ldots - 2018^2 + 2019^2\\).\n\nRemember to put your answer on its o... | 2039190 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed ci... | 622 |
[
{
"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 maximum integer $n$ such that for each positive integer $k \\le \\frac{n}{2}$, there exist t... | 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 $Answer is the answer to the problem.\n\nThree students write on the blackboard three two-digit square numbers next to each other. In the end, they... | 166464 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $x$ if \\[ \\frac{1}{\\frac{1}{\\frac{1}{\\frac{1}{x} + \\frac{1}{2}} + \\frac{1}{\\frac{1}{x} + \\fr... | 70 |
[
{
"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 $m$ and $n$ be positive integers such that $x = m + \\sqrt{n}$ is a solution to the equation $x^2 - 10... | 55 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of positive integral values of $p$ for which there exists a triangle with sides $a$, ... | 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 $Answer is the answer to the problem.\n\nIn a circle, 15 equally spaced points are drawn, and arbitrary triangles are formed by connecting 3 of the... | 19 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe following figure shows a walk of length 6:\n\n```asy\nunitsize(20);\nfor (int x = -5; x <= 5; ++x)\n ... | 239 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGraphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these citi... | 1005 |
[
{
"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\nEvaluate $2023 \\cdot \\frac{2023^6 + 27}{(2023^2 + 3)(2024^3 - 1)} - 2023^2$.\n\nRemember to put your ans... | -6066 |
[
{
"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 an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at ra... | 628 |
[
{
"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 $x, y$ be positive real numbers. If \\[129-x^2=195-y^2=xy,\\] then $x = \\frac{m}{n}$ for relatively p... | 4306 |
[
{
"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 $Answer is the answer to the problem.\n\nFind the number of pairs \\((n, C)\\) of positive integers such that \\(C \\leq 100\\) and \\(n^2 + n + C\... | 180 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1 < a_2 < a_3 < \\ldots < a_n < \\ldots$ be positive integers such that, for $n = 1, 2, 3, \\ldots,... | 2014 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt Ignus School, there are $425$ students. Of these students, $351$ study mathematics, $71$ study Latin, 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\nFour rectangular strips, each measuring $4$ by $16$ inches, are laid out with two vertical strips crossing... | 96 |
[
{
"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 square of the table below, we must write a different integer from $1$ to $17$, such that the sum o... | 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\nWhen flipped, coin A shows heads $\\frac{1}{3}$ of the time, coin B shows heads $\\frac{1}{2}$ of the time... | 273 |
[
{
"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\nWe wish to find the sum of $40$ given numbers using $40$ processors. Initially, each processor displays th... | 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\nOn the planet Mars, there are $100$ states that are in dispute. To achieve a peace situation, blocs must b... | 6 |
[
{
"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 $Answer is the answer to the problem.\n\nIf\n\\[\n\\sum_{k=1}^{1000}\\left( \\frac{k+1}{k}+\\frac{k}{k+1}\\right)=\\frac{m}{n}\n\\]\nfor relatively... | 2004001 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (i... | 27 |
[
{
"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 $a_1, a_2, \\ldots, a_{18}$ be a list of prime numbers such that $\\frac{1}{64} \\times a_1 \\times a_... | 64008420 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ... | 25 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with ... | 2128 |
[
{
"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 $Answer is the answer to the problem.\n\nFind the decimal form of the largest prime divisor of \\(100111011_6\\).\n\nRemember to put your answer on... | 181 |
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{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that... | 1010 |
[
{
"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\nCalculate $\\sum_{i=2}^{\\infty} \\frac{i^2-2}{i!}$.\n\nRemember to put your answer on its own line after ... | 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\nFor any interval $\\mathcal{A}$ on the real number line not containing zero, define its reciprocal to be t... | 60 |
[
{
"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 $n$ be an arbitrary positive integer. Calculate \\( S_n = \\sum_{r=0}^n 2^{r-2n} \\binom{2n-r}{n} \\).... | 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\nThe triangle $ABC$ is isosceles with $AB = BC$. The point $F$ on the side $[BC]$ and the point $D$ on the ... | 108 |
[
{
"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 each of the numbers $n + 1$, $n + 3$, $n + 4$, $... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor what single digit $n$ does $91$ divide the 9-digit number $12345n789$?\n\nRemember to put your answer ... | 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\nIn trapezoid $ABCD$, $AB < CD$, $AB \\perp BC$, $AB \\parallel CD$, and the diagonals $AC$, $BD$ are perpe... | 61 |
[
{
"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 $24$-hour digital clock shows times $h : m : s$, where $h$, $m$, and $s$ are integers with $0 \\leq h \\... | 1164 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be the product of the first $50$ nonzero square numbers. Find the largest integer $k$ such that $7... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake... | 95 |
[
{
"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 positive integers $n$ where the mean and median of \\( \\{20, 42, 69, n\\} \\) are bot... | 45 |
[
{
"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 positive integers $n$ satisfying the inequalities\n\\[\n2^{n-1} < 5^{n-3} < 3^n.\n\\... | 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\nFind all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that $f(x + y) = f(x) + f(y) + f(xy)$ for all $x... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe letters $R$, $M$, and $O$ represent whole numbers. If the following conditions are satisfied: \n1. $R ... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a positive integer $N \\ge 2$ \"special\" if for every $k$ such that $2 \\leq k \\leq N$, $N$ can be ... | 50 |
[
{
"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 $x$ which satisfies the following congruences:\n\n1. $x^2 - 8x + 1 \... | 35 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many subsets with $3$ elements of the set $S = \\{1, 2, 3, \\ldots, 19, 20\\}$ exist, such that their ... | 795 |
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