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\nDetermine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two gi... | 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\nLet $f$ be a function from the set $X = \\{1,2, \\dots, 10\\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for... | 961 |
[
{
"content": "Solve the following math problem step by 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 natural number $n$ such that, if the set $\\{1,2, \\ldots, n\\}$ is arbitrarily divided into two non-intersecti... | 96 |
[
{
"content": "Solve the following math problem step by 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 $m^4 - n^4 = 3439$. What is the value of $mn$?\n\nRemember to put your answer o... | 90 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many sequences of real numbers $a_1, a_2, \\ldots, a_9$ satisfy \n\\[ |a_1 - 1| = |a_2 - a_1| = \\cdots = |a_9 - a_8| = |1... | 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\nLet $p=2017$ be a prime. Find the remainder when \\[ \\left\\lfloor \\frac{1^p}{p} \\right\\rfloor + \\left\\lfloor \\frac{2^p... | 1010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nStart with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubl... | 344 |
[
{
"content": "Solve the following math problem step by 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 $C$ denote the curve $y^2 = \\frac{x(x+1)(2x+1)}{6}$. The points $(\\frac{1}{2}, a)$, $(b, c)$, and $(24, d)$ lie on $C$ a... | 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\nLet $a$ and $b$ be nonzero real numbers such that $\\frac{1}{3a} + \\frac{1}{b} = 2011$ and $\\frac{1}{a} + \\frac{1}{3b} = 1$... | 1509 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting tw... | 9999 |
[
{
"content": "Solve the following math problem step by 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 $P(x)$ be a real quadratic trinomial such that for all $x \\in \\mathbb{R}$, the inequality $P(x^3 + x) \\geq P(x^2 + 1)$ ... | 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\nAs in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ ... | 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\nThe sequence of nonnegative integers $F_0, F_1, F_2, \\dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_... | 638 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\\overline{BC}$ and $\\overline{AB}$, re... | 91 |
[
{
"content": "Solve the following math problem step by 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 \\textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules:\n\\begin{itemize}\n ... | 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 $f(n)$ denote the sum of the distinct positive integer divisors of $n$. Evaluate:\n\\[\nf(1) + f(2) + f(3) + f(4) + f(5) +... | 69 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAfter Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle... | 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 $Answer is the answer to the problem.\n\nWhat is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bis... | 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 $Answer is the answer to the problem.\n\nFind the three-digit number that has the greatest number of different divisors.\n\nRemember to put your answer on its own line... | 840 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \\[ \\text{lcm}(m,n) = 3m... | 972 |
[
{
"content": "Solve the following math problem step by 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 rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest... | 311 |
[
{
"content": "Solve the following math problem step by 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 $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression:\n\\[\n(r^2 + s + t)(s^2 ... | 1100 |
[
{
"content": "Solve the following math problem step by 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 integer solutions to $||x| - 2020| < 5$.\n\nRemember to put your answer on its own line after \"Answer:\"."... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last eight digits of the binary representation of $27^{1986}$.\n\nRemember to put your answer on its own line after \... | 11011001 |
[
{
"content": "Solve the following math problem step by 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 the equation $x^5 - 180x^4 + Ax^3 + Bx^2 + Cx + D = 0$ are in geometric progression. The sum of their reciprocals... | 243 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe figure on the right shows the ellipse \\( \\frac{(x-19)^2}{19} + \\frac{(y-98)^2}{98} = 1998 \\).\n\nLet \\( R_1, R_2, R_3... | 7448 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many different orders can the characters $P \\ U \\ M \\ \\alpha \\ C$ be arranged such that the $M$ is to the left of ... | 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\nKevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign er... | 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 $ \\lim_{x\\rightarrow 0} \\frac{1}{x} \\int_{-x}^x (t\\sin 2006 t + 2007t + 1004)\\ dt$.\n\nRemember to put your answer ... | 2008 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppo... | 75 |
[
{
"content": "Solve the following math problem step by 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 nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number.\n\nRemember to put your answer... | 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\nPositive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.\n\nRemember to put your a... | 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\nFind the remainder when $2^{5^9} + 5^{9^2} + 9^{2^5}$ is divided by $11$.\n\nRemember to put your answer on its own line after... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\\angle BAC$ intersect line $BC$ at $D$. If there e... | 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 $Answer is the answer to the problem.\n\nFind the numerator of \\[ \\frac{1010\\overbrace{11 \\ldots 11}^{2011 \\text{ ones}}0101}{1100\\underbrace{11 \\ldots 11}_{201... | 9091 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \\times 100$ rectang... | 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\nA building contractor needs to pay his $108$ workers \\(\\$200\\) each. He is carrying $122$ one hundred dollar bills and $188... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive... | 1847 |
[
{
"content": "Solve the following math problem step by 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 $\\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\\mathc... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of ordered pairs of positive integers $(x, y)$ with $y < x \\le 100$ such that $x^2 - y^2$ and $x^3 - y^3... | 99 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $n$ such that \\( \\frac{2020}{n^3 + n} \\) is an integer.\n\nRemember to put your answe... | 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\nWhat is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any tw... | 672 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose two equally strong tennis players play against each other until one player wins three games in a row. The results of e... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\\mathbb{N... | 10201 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of the cells of a $7 \\times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed co... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $27,000,001$ has exactly four prime factors. Find the sum of these prime factors.\n\nRemember to put your answer on... | 652 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, where $\\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\\angle ABD$ is a right angle. ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions... | 47 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that \\( \\left\\lfloor \\frac{n}{2} \\right\\rfloor \\cdot \\left\\lfloor \\frac{n}{3} \\... | 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 $Answer is the answer to the problem.\n\nHow many ordered pairs of integers \\((m,n)\\) are there such that \\(m\\) and \\(n\\) are the legs of a right triangle with a... | 87 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1, a_2, \\ldots$ be a sequence defined by $a_1 = 1$ and for $n \\ge 1$, $a_{n+1} = \\sqrt{a_n^2 - 2a_n + 3} + 1$. Find ... | 33 |
[
{
"content": "Solve the following math problem step by 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 $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are e... | 628 |
[
{
"content": "Solve the following math problem step by 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 $19^{92}$ is divided by 92.\n\nRemember to put your answer on its own line after \"Answer:\".",
"r... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than fi... | 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\nSuppose $ABCD$ is a trapezoid with $AB \\parallel CD$ and $AB \\perp BC$. Let $X$ be a point on segment $\\overline{AD}$ such ... | 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\nConvex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\\angle ABC = 90^\\circ$. Point $P$ is on $DA$ such that $\\an... | 630 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAnn and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConstruction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, ... | 51 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $x, y \\in \\mathbb{Z}$ satisfy the equation:\n\\[\ny^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 +... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than $200$ are relatively prime to either $15$ or $24$?\n\nRemember to put your answer on its ... | 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\nLet $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that \\(\\text{area} (\\triangle AD... | 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\nLet $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\\overline{abcd}$, where $a \\neq 0$. ... | 9376 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \\ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor which natural number $k$ does \\( \\frac{k^2}{1.001^k} \\) attain its maximum value?\n\nRemember to put your answer on its... | 2001 |
[
{
"content": "Solve the following math problem step by 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 integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the ... | 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\nFind the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements ... | 2017 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered pairs of real numbers $(x, y)$ are there such that $x^2 + y^2 = 200$ and \\[ \\sqrt{(x-5)^2 + (y-5)^2} + \\sq... | 12 |
[
{
"content": "Solve the following math problem step by 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 an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ ... | 10087 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ordered pairs of non-negative integers \\((x, y)\\) which satisfy \\(x^2 + y^2 = 32045.\\)\n\nRemember t... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $2023 \\times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For every dist... | 1022121 |
[
{
"content": "Solve the following math problem step by 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 $S=\\left\\{ \\frac{p}{q} \\mid q \\leq 2009, \\frac{p}{q} < \\frac{1257}{2009}, p,q \\in \\mathbb{N} \\right\\}$. If the ... | 595 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face ... | 1195 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a certain country, there are $2023$ islands and $2022$ bridges. Each bridge connects two different islands, and any two isl... | 1011 |
[
{
"content": "Solve the following math problem step by 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 two positive integers $x, y$ satisfy the condition $44 \\mid (x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$... | 21296 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAndy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registe... | 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\nThe two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $... | 683 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants t... | 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 $Answer is the answer to the problem.\n\nIncluding the original, how many ways are there to rearrange the letters in \"PRINCETON\" so that no two vowels \\((I, E, O)\\... | 17280 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, \\ldots, r_{10}$, and let $Q$ be a $45$-degree monic polynomia... | 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 $Answer is the answer to the problem.\n\nThe number of positive integer pairs \\((a,b)\\) such that \\(a\\) divides \\(b\\) and \\(b\\) divides \\(2013^{2014}\\) can b... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectiv... | 200 |
[
{
"content": "Solve the following math problem step by 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 complex number, and set $\\alpha$, $\\beta$, and $\\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$... | 2009 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFarmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as p... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $A_1, A_2, A_3, \\ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be form... | 640 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\\sqrt{k}$ and let $b_k$ be the greatest integ... | 40842 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a) = b + c$, ... | 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\nIn a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\\alpha$, $\\beta$, and $\\alpha + \\beta$ ra... | 49 |
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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 integers between $2$ and $100$ have only odd numbers in their prime factorizations?\n\nRemember to put your answer on... | 49 |
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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 hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the... | 216 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be a set. We say $S$ is $D^\\ast$-finite if there exists a function $f : S \\to S$ such that for every nonempty proper... | 120 |
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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 $Answer is the answer to the problem.\n\nJeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy ... | 87 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m - 3n)^2 = 2019$. Find $3m + n$.\n\nRemember to put your answer... | 46 |
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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\nLine $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$.\n\... | 50 |
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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 $y_0$ be chosen randomly from $\\{0, 50\\}$, let $y_1$ be chosen randomly from $\\{40, 60, 80\\}$, let $y_2$ be chosen ran... | 107 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A, B, C, \\ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \\[ \\l... | 5 |
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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 the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \\dots, S_{2015})$ of $\\{1, 2, \\dot... | 2030612 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA canister contains $2\\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\\frac{1}{3}$ cups of... | 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\nIn triangle $ABC$, where $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\\angle BAD = \\... | 18 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that ... | 5 |
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