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\nA positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \\ldots$. The smallest p... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvery one of the six trucks of a construction company drove for $8$ hours, and they all together spent $720$ litres of oil. Ho... | 810 |
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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 for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative ... | 2021 |
[
{
"content": "Solve the following math problem step by 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\\textbf{Problem 1.} There are less than $400$ marbles. If they are distributed among $3$ children, there is one left over. If... | 310 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a Pacman of radius $1$, and a mouth opening angle of $90^\\circ$, what is the largest circular pellet it can eat? The pe... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider real numbers $A$, $B$, \\dots, $Z$ such that:\n\\[\\begin{align*}\n EVIL &= \\frac{5}{31}, \\\\\n LOVE &= \\frac{6}... | 579 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence of letters **TAGC** is written in succession 55 times on a strip, as shown below. The strip is to be cut into seg... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings s... | 510 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. F... | 86 |
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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\nTriangle $ABC$ has side lengths $AB=75$, $BC=100$, and $CA=125$. Point $D$ is the foot of the altitude from $B$, and $E$ lies ... | 864 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:\n\n1. The sum of the f... | 200 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.\n\nRemember to put your answer on its own line af... | 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 $Answer is the answer to the problem.\n\nSixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divi... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\\angle AEB$, $\\angle BF... | 98 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSteven the troll submits problems using silly names instead of his own. On day $1$, he gives no name at all. Every subsequent ... | 16 |
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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\nTerry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps... | 59 |
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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\nAround a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their difference ... | 1005 |
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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 $Answer is the answer to the problem.\n\nThere are several pairs of integers \\((a, b)\\) satisfying the equation:\n\\[ a^2 - 4a + b^2 - 8b = 30 \\]\nFind the sum of t... | 72 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is know... | 504 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the future, each country in the world produces its Olympic athletes via cloning and strict training programs. Therefore, in... | 2520 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive three-digit integers $abc$ can represent a valid date in $2013$, where either $a$ corresponds to a month and... | 273 |
[
{
"content": "Solve the following math problem step by 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 integer $k$ such that:\n\\[\nk \\leq \\sqrt{2} + \\sqrt[3]{\\frac{3}{2}} + \\sqrt[4]{\\frac{4}{3}} + \\sqrt[5... | 2014 |
[
{
"content": "Solve the following math problem step by 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 an isosceles triangle $\\triangle ABC$ where $AB = BC$. A point $K$ is inside $\\triangle ABC$ such that $CK = AB = BC$ ... | 150 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8, 15, 8, 8, 8, 6, 8,$ and $29$ as shown. All... | 80 |
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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 maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is an exact square.\n\nRemember to put your answer on ... | 1972 |
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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 $f(x) = x + \\frac{1}{2x + \\frac{1}{2x + \\frac{1}{2x + \\cdots}}}$. Find $f(99)f'(99)$.\n\nRemember to put your answer o... | 99 |
[
{
"content": "Solve the following math problem step by 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 geometric sequence $a_1,a_2,a_3,\\dots$ satisfies $a_1=1$ and \\[\\frac{1}{a_1a_2}+\\frac{1}{a_2a_3}+\\frac{1}{a_3... | 102 |
[
{
"content": "Solve the following math problem step by 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 ship *Meridiano do Bacalhau* conducts its fishing business over a period of $64$ days. Each day, the captain chooses a dir... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be di... | 247 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : ... | 80 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green mar... | 323 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ rig... | 100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the ... | 2224 |
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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 concert, $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform rig... | 38 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$P(x)$ is a polynomial of degree $3n$ such that:\n\n\\[\n\\begin{align*}\nP(0) = P(3) = \\cdots &= P(3n) = 2, \\\\\nP(1) = P(4... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $X = \\{1, 2, \\ldots, 2017\\}$. Let $k$ be a positive integer. Given any $r$ such that $1 \\le r \\le k$, there exist $k$... | 2017 |
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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 meeting of $12k$ people, each person exchanges greetings with exactly $3k + 6$ others. For any two people, the number of ... | 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\nReimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as ... | 1727 |
[
{
"content": "Solve the following math problem step by 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 trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\\frac{1}{3}$, and the slope of line $A... | 1706 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute \\[\\sum_{n_{60}=0}^2\\sum_{n_{59}=0}^{n_{60}}\\cdots\\sum_{n_2=0}^{n_3}\\sum_{n_1=0}^{n_2}\\sum_{n_0=0}^{n_1}1.\\]\n\... | 1953 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAll of the roots of $x^3 + ax^2 + bx + c$ are positive integers greater than 2, and the coefficients satisfy $a + b + c + 1 = ... | -58 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.\n\nRemember to put your answer on ... | 85783 |
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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 meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each other, among the remaining people, ... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer is called *cool* if it can be expressed in the form $a! \\cdot b! + 315$ where $a$ and $b$ are positive int... | 317 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of ... | 606 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhich is the largest positive integer $n$ satisfying the inequality: $n^{2007} > (2007)^n$?\n\nRemember to put your answer on ... | 2006 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be an equilateral triangle and $P$ a point on $\\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.\n\n... | 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 $Answer is the answer to the problem.\n\nA dartboard consists of three concentric circles with radii 4, 6, and 8. Three darts are thrown at the board, sticking at rand... | 617 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\\angle MAC = \\angle ABC$ and $\\angle BAM =... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the numbers $x_1, x_2, \\ldots, x_{101}$ is $\\pm 1$. What is the smallest positive value of $\\sum_{1\\leq i < j \\le... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}$ and $20^{3... | 2301 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N_{15}$ be the answer to problem 15.\n\nFor a positive integer $x$ expressed in base ten, let $x'$ be the result of swapp... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that both the sum $n + 10$ and the product $n \\times 10$ are perfect squares.\n\n... | 90 |
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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 $Answer is the answer to the problem.\n\nWendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular ta... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value, and your score i... | 168 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSam spends his days walking around the following $2 \\times 2$ grid of squares:\n\\[\n\\begin{array}{|c|c|}\n\\hline\n1 & 2 \\... | 167 |
[
{
"content": "Solve the following math problem step by 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 real number $a$, the floor of $a$, written $\\lfloor a \\rfloor$, is the greatest integer less than or equal to $a$. F... | 362 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhile entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$... | 941 |
[
{
"content": "Solve the following math problem step by 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 natural numbers $1 \\leq n \\leq 10^6$ such that the least prime divisor of $n$ is $17$.\n\nRemember to ... | 11284 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many pairs of positive integers \\((a, b)\\) satisfy the equation \\(\\log_a 16 = b\\)?\n\nRemember to put your answer on ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?\n\nRemember to put your answ... | 1033 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many natural numbers exist such that their square results in a thirty-digit number with the following property: When divid... | 18377 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be the two possible values of $\\tan\\theta$ given that \\[\\sin\\theta + \\cos\\theta = \\frac{193}{137}.\\] ... | 28009 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers $n$ between $1$ and $2021$ does the infinite nested expression $\\sqrt{n + \\sqrt{n + \\sqrt{n + \\sqrt{... | 44 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn trapezoid $ABCD$, where $BC \\parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.\n\nRem... | 252 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest integer $n$ for which $\\frac{10!}{n}$ is a perfect square?\n\nRemember to put your answer on its own lin... | 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\nThere are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers wan... | 2013 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$ be a positive integer which is a multiple of 3 but not a multiple of 9. If adding the product of each digit of $A$ to ... | 138 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nStudents are in a classroom with $n$ rows. In each row, there are $m$ tables. It is given that $m, n \\geq 3$. At each table, ... | 153 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the integer part of \\( \\sum_{n=1}^{40000} \\frac{1}{\\sqrt{n}} \\).\n\nRemember to put your answer on its own line afte... | 398 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a regular $17$-gon $\\mathcal{P}$ and its circumcircle $\\mathcal{Y}$ on the plane. The vertices of $\\mathcal{P}$ ar... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor every $a \\in \\mathbb{N}$, denote by $M(a)$ the number of elements in the set\n\\[ \\{ b \\in \\mathbb{N} \\mid a + b \\t... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the total number of different integer values the function \n\\[ f(x) = \\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lef... | 734 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nI have a $2 \\times 4$ grid of squares. How many ways can I shade at least one of the squares such that no two shaded squares ... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $0.3010 < \\log 2 < 0.3011$ and $0.4771 < \\log 3 < 0.4772$, find the leftmost digit of $12^{37}$.\n\nRemember to p... | 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\nWhat is the greatest common divisor of the set of numbers $\\{16^n + 10n - 1 \\mid n=1,2,\\ldots \\}$?\n\nRemember to put your... | 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\nFor a positive integer $n$, let $s(n)$ be the sum of the digits of $n$ when $n$ is expressed in base ten. For example, $s(2022... | 4107 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f_1(n)$ be the number of divisors of $n$, and define $f_k(n) = f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such th... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA permutation \\((a_1, a_2, a_3, \\dots, a_{2012})\\) of \\((1, 2, 3, \\dots, 2012)\\) is selected at random. If \\(S\\) is th... | 2083 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor integers $0 \\le m,n \\le 64$, let $\\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\\left\\lfloor \\fr... | 208 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAna and Banana play a game. First, Ana picks a real number $p$ with $0 \\le p \\le 1$. Then, Banana picks an integer $h$ great... | 382 |
[
{
"content": "Solve the following math problem step by 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 Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and... | 4293 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the... | 1200 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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 li... | 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 $Answer is the answer to the problem.\n\nEvaluate the expression:\n\\[2^{7} \\times 3^{0} + 2^{6} \\times 3^{1} + 2^{5} \\times 3^{2} + \\cdots + 2^{0} \\times 3^{7}.\... | 6305 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA lucky number is a number whose digits are only $4$ or $7$. What is the $17^{th}$ smallest lucky number?\n\nRemember to put y... | 4474 |
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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\nNatural numbers $k, l, p$, and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$, then $a + \\frac{1}{b}$ and $... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMaria has $14$ days to train for an olympiad. The conditions are:\n1. She cannot train for $3$ consecutive days.\n2. She canno... | 1220 |
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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 square $BCDE$ is inscribed in circle $\\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A \"hook\" is... | 57 |
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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 point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered ... | 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 $Answer is the answer to the problem.\n\nMisha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transacti... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\\theta$ be the measure of the a... | 55 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe subnumbers of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For examp... | 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 $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of... | 8 |
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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 $ABCD$ and $AEFG$ be unit squares such that the area of their intersection is $\\frac{20}{21}$. Given that $\\angle BAE < ... | 4940 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRectangle $ABCD$ measures $70$ by $40$. Eighteen points (including $A$ and $C$) are marked on the diagonal $AC$, dividing the ... | 600 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y + \\frac{349}{4} = 0$. Fin... | 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\nLet $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $... | 444 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given that $RI=3$... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_1, x_2, \\dots, x_{100}$ be real numbers such that $|x_1| = 63$ and $|x_{n+1}| = |x_n + 1|$ for $n = 1, 2, \\dots, 99$.... | 2034 |
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