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\nPositive integers $a$, $b$, and $c$ satisfy the equation:\n\\[\n\\frac{ab}{a - b} = c\n\\]\nWhat is the largest possible value... | 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\nFind the number of 5-digit numbers $n$, such that every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by ... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a square with side length $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, ... | 120 |
[
{
"content": "Solve the following math problem step by 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 least positive integer $m$ such that $\\text{lcm}(15, m) = \\text{lcm}(42, m)$. Here $\\text{lcm}(a, b)$ is the least... | 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\nDenote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi... | 68 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $k$ such that \n\\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\\... | 1297 |
[
{
"content": "Solve the following math problem step by 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 polynomial such that, for all real numbers $x$, \\( f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019 \\).\nCompu... | -2019 |
[
{
"content": "Solve the following math problem step by 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 game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are given a rectangular playing field of size $13 \\times 2$ and any number of dominoes of sizes $2 \\times 1$ and $3 \\ti... | 257 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $M$, $N$, $P$ are selected on sides $\\overline{AB}$, $\\overline{AC}$, $\\overline{BC}$, respectively, of triangle $AB... | 150 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers have a square less than $10^7$?\n\nRemember to put your answer on its own line after \"Answer:\".",... | 3162 |
[
{
"content": "Solve the following math problem step by 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 square grid $100 \\times 100$ is tiled in two ways: only with dominoes and only with $2 \\times 2$ squares. What is the leas... | 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 $Answer is the answer to the problem.\n\nWhat is the largest prime factor of 8091?\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "use... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $k$ be an integer. If the equation $(x-1)\\left|x+1\\right|=x+\\frac{k}{2020}$ has three distinct real roots, how many dif... | 4544 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $7^{7^7}$ is divided by $1000$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 343 |
[
{
"content": "Solve the following math problem step by 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 trapezium in which $AB \\parallel CD$ and $AD \\perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at... | 84 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRecall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \\leq a <... | 98 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the min... | 448 |
[
{
"content": "Solve the following math problem step by 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 $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = \\cdots = f(x_n) = 2002$... | 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 $S$ be the set of all natural numbers with the property: the sum of the largest three divisors of a number $n$, excluding ... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited numb... | 252 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn $n$-sided regular polygon with side length $1$ is rotated by $\\frac{180^\\circ}{n}$ about its center. The intersection poi... | 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\nThe product \n$$\\frac{1}{2} \\cdot \\frac{2}{4} \\cdot \\frac{3}{8} \\cdot \\frac{4}{16} \\cdot \\ldots \\cdot \\frac{99}{2^{... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023... | 783 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSay that a polynomial with real coefficients in two variables, $x, y$, is balanced if the average value of the polynomial on e... | 2020050 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n480 one-centimeter unit cubes are used to build a block measuring 6 cm by 8 cm by 10 cm. A tiny ant then chews its way in a st... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\phi = \\frac{1+\\sqrt{5}}{2}$. A base-$\\phi$ number $(a_n a_{n-1} \\ldots a_1 a_0)_{\\phi}$, where $0 \\le a_n, a_{n-1... | 1048576 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one pl... | 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 $Answer is the answer to the problem.\n\nA *form* is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all ... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least positive integer $k$ such that when $\\frac{k}{2023}$ is written in simplest form, the sum of the numerator and... | 35 |
[
{
"content": "Solve the following math problem step by 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 $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.\n\nRemember to put your a... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSiva has the following expression, which is missing operations:\n$$\\frac{1}{2} \\_ \\frac{1}{4} \\_ \\frac{1}{8} \\_ \\frac{1... | 39 |
[
{
"content": "Solve the following math problem step by 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 Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 2... | 142 |
[
{
"content": "Solve the following math problem step by 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 square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is co... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d$ be real numbers satisfying the system of equations:\n\\[\n\\begin{align*}\n(a+b)(c+d) &= 2, \\\\\n(a+c)(b+d) ... | 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\nLet $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all four-digit natural numbers $\\overline{xyzw}$ such that the sum of the number and the sum of its digits equals $2003$... | 1978 |
[
{
"content": "Solve the following math problem step by 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$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only co... | 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\nLet $\\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a ... | 197 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCaroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot t... | 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\nIn the parallelogram $ABCD$, we are given that $AC = 10$ and $BD = 28$. The points $K$ and $L$ are such that they move in the ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( a\\uparrow\\uparrow b = a^{a^{\\cdots^{a}}} \\), where there are \\( b \\) instances of \\( a \\). The operation \\( a... | 27 |
[
{
"content": "Solve the following math problem step by 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 plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following co... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFarmer John has $5$ cows, $4$ pigs, and $7$ horses. How many ways can he pair up the animals so that every pair consists of an... | 100800 |
[
{
"content": "Solve the following math problem step by 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 $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ han... | 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\nCompute how many permutations of the numbers $1, 2, \\dots, 8$ have no adjacent numbers that sum to $9$.\n\nRemember to put yo... | 13824 |
[
{
"content": "Solve the following math problem step by 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 number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2... | 494 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWilliam is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ has side lengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with si... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne integer was removed from the set $S=\\{1, 2, 3, \\ldots, n\\}$ of the integers from $1$ to $n$. The arithmetic mean of the... | 61 |
[
{
"content": "Solve the following math problem step by 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 solution to the equation $2^{2^k} \\equiv k \\pmod{29}$.\n\nRemember to put your answer on ... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $r$ be a real number such that $\\sqrt[3]{r} - \\frac{1}{\\sqrt[3]{r}} = 2$. Find $r^3 - \\frac{1}{r^3}$.\n\nRemember to p... | 2786 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?\n\nRemember to put your ... | 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\nCompute the sum of the distinct prime factors of $10101$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$, \n\n$$a_{n+1} = \\frac{\\... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to arrange the array of positive integers such that:\n- The number $2012$ is in the first position.\n-... | 201 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers \\((a, b)\\) that satisfy the inequality\n\n\\[\n1 < a < b+2 < 10.\n\\]\n\nRememb... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves ... | 280 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$q$ is the smallest rational number having the following properties:\n\n1. $q > \\frac{31}{17}$\n\n2. When $q$ is written in i... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the $4 \\times 4$ array of $16$ dots, shown below.\n\nCounting the number of squares whose vertices are among the $16... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is gre... | 101 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPortraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than ... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) has \\( AB = 4 \\) and \\( AC = 6 \\). Let point \\( D \\) be on line \\( AB \\) so that \\( A \\) is b... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so t... | 3616 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all four-digit numbers $\\overline{abcd}$ such that\n\n$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^... | 2010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$, and $d$ be positive integers such that $77^a \\cdot 637^b = 143^c \\cdot 49^d$. Compute the minimal value o... | 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\nPassing through the origin of the coordinate plane are $180$ lines, including the coordinate axes, which form $1^\\circ$ angle... | 8950 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ordered quadruples $(a, b, c, d)$ of distinct positive integers such that $\\binom{\\binom{a}{b}}{\\bino... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many two-digit numbers have exactly $4$ positive factors? (Here $1$ and the number $n$ are also considered as factors of $... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy:\n\\[\n\\begin{array}{c@{\\,}c@{\\,}c@{\\,}c}\n & C & C & ... | 42 |
[
{
"content": "Solve the following math problem step by 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_{i=0}^{1000} \\frac{\\binom{1000}{i}}{\\binom{2019}{i}}\\] can be expressed in the form $\\frac{p}{q}$, where... | 152 |
[
{
"content": "Solve the following math problem step by 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 $P(x)$ is the least degree polynomial with rational coefficients such that \n\\[P(\\sqrt{2} + \\sqrt{3}) = \\sqrt{2... | 455 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDima has written the numbers $\\frac{1}{80!}, \\frac{1}{81!}, \\ldots, \\frac{1}{99!}$ on 20 infinite pieces of paper as decim... | 155 |
[
{
"content": "Solve the following math problem step by 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(x) = x^{2008} + x^{2007} + x^{2006} + \\cdots + x + 1, \\] and let $r(x)$ be the polynomial remainder when $p(x)$ is... | 64 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to ... | 149 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe call $\\overline{a_n\\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \n\\[k = \\sum_{i=2}^n a_i F_i,... | 1596 |
[
{
"content": "Solve the following math problem step by 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 three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\\%$ solu... | 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\nA container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ cent... | 60 |
[
{
"content": "Solve the following math problem step by 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 $a_1, a_2, \\ldots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Dete... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\([n] = \\{1, 2, 3, \\ldots, n\\}\\) and for any set \\(S\\), let \\(P(S)\\) be the set of non-empty subsets of \\(S\\). ... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $A = 3^3 \\cdot 33^{33} \\cdot 333^{333} \\cdot 3333^{3333}$ is divided by $100$.\n\nRemember to put y... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at rand... | 58 |
[
{
"content": "Solve the following math problem step by 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 two prime numbers $p$ such that $5p$ can be expressed in the form $\\left\\lfloor \\dfrac{n^2}{5} \\right\\rfloor$ f... | 52 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nScrews are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence, each of t... | 60 |
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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 $Answer is the answer to the problem.\n\nCarol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she mult... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA student at Harvard named Kevin was counting his stones by $11$. He messed up $n$ times and instead counted by $9$s, and woun... | 21 |
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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 $a$ be a positive real number such that \\(4a^2+\\frac{1}{a^2}=117.\\) Find \\(8a^3+\\frac{1}{a^3}.\\)\n\nRemember to put ... | 1265 |
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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 $S_n = n^2 + 20n + 12$, where $n$ is a positive integer. Determine the sum of all possible values of $n$ for which $S_n$ i... | 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\nXonathan Jue goes to the casino with exactly $1000. Each week, he has a $\\frac{1}{3}$ chance of breaking even and a $\\frac{2... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of two-digit positive integers that are divisible by both of their digits. For example, $36$ is one of thes... | 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\nFrankie the Frog starts his morning at the origin in $\\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered triples $\\left(a, b, c\\right)$ of positive integers are there such that at least two of $a, b, c$ are prime... | 30 |
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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 shipping company charges $0.30l + 0.40w + 0.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l,... | 276 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \\frac{1}{n^2}$ for $n = 1, 2, \\dots, 2016$. Find $\\lfloor ... | -9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that \\[ \\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\\cdots+a_{m}x^{k_{m}} \\] where $a_{1}$,... | 665280 |
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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 greatest positive integer $n$ such that $2^n$ divides \\(\\text{lcm}(1^1, 2^2, 3^3, \\ldots, 2016^{2016})\\).\n\nReme... | 14336 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $C$ and $D$ be points inside angle $\\angle AOB$ such that $5\\angle COD = 4\\angle AOC$ and $3\\angle COD = 2\\angle DOB$... | 28 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ between $1$ and $N = 2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers s... | 1109 |
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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 minimum value of $2x^2 + 2y^2 + 5z^2 - 2xy - 4yz - 4x - 2z + 15$ for real numbers $x$, $y$, $z$.\n\nRemember to put y... | 10 |
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