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\nTwo people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different suits. At the beginning,... | 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\nIn the quadrilateral $ABCD$, angles $\\angle B$ and $\\angle C$ are equal to $120^\\circ$. Given $AB = CD = 1$ and $CB = 4$, f... | 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 $Answer is the answer to the problem.\n\nSix children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. The... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive... | 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 $S = 2^3 + 3^4 + 5^4 + 7^4 + \\cdots + 17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find th... | 93 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBob chooses a $4$-digit binary string uniformly at random and examines an infinite sequence of uniformly and independently ran... | 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 $w, x, y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions ho... | 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\nThere are seven cards in a hat, and on card $k$ there is a number $2^{k-1}$, where $k=1,2,...,7$. Solarin picks the cards at r... | 127 |
[
{
"content": "Solve the following math problem step by 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 $f$ is a monic cubic polynomial with $f(0) = -64$, and all roots of $f$ are non-negative real numbers, what is the largest ... | -125 |
[
{
"content": "Solve the following math problem step by 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 rectangular grid with side lengths that are integers greater than $1$ is given. Smaller rectangles, each with an area equal ... | 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\nHow many positive integers $N$ satisfy all of the following three conditions?\n\n1. $N$ is divisible by $2020$.\n2. $N$ has at... | 508536 |
[
{
"content": "Solve the following math problem step by 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 subset of $\\{1, 2, \\dots , 1000000\\}$ such that for any $x, y \\in A$ with $x \\neq y$, we have $xy \\notin A$... | 999001 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the remainder when $7^{8^9}$ is divided by $1000$?\n\nRemember to put your answer on its own line after \"Answer:\".",... | 801 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, and $c$ be positive integers satisfying the following equations:\n\n$$a^4 + a^2b^2 + b^4 = 9633$$\n$$2a^2 + a^2b... | 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\nPoints $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A, E, C$ lie on a line in that order, $B, E, D$ lie on a l... | 967 |
[
{
"content": "Solve the following math problem step by 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 $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$.\n\nRemember to put your answer on its own line after \"... | 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\nLet $P(A)$ be the arithmetic mean of all elements of the set $A = \\{ a_1, a_2, \\ldots, a_n \\}$, defined as $P(A) = \\frac{1... | 51 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is an isosceles triangle $\\triangle ABC$ with $\\overline{AB} = \\overline{AC} = 5$ and $\\overline{BC} = 6$. Points $D... | 72 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe following diagram shows an equilateral triangle $\\triangle ABC$ and three other triangles congruent to it. The other thre... | 768 |
[
{
"content": "Solve the following math problem step by 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 set of natural numbers. Suppose $f: N \\to N$ is a function satisfying the following conditions: \n1. $f(mn) =... | 210 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIntegers $a$, $b$, and $c$ satisfy the equations $a + b - c = 1$ and $a^2 + b^2 - c^2 = -1$. Determine the sum of all possible... | 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\nNon-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\\angle ABC = \... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.\n\nRem... | 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 $Answer is the answer to the problem.\n\nThere are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a stra... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$... | 41 |
[
{
"content": "Solve the following math problem step by 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 numbers $\\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with... | 162 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIntegers $1, 2, 3, \\ldots, n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are... | 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\nSolve in positive real numbers: $n + \\lfloor \\sqrt{n} \\rfloor + \\lfloor \\sqrt[3]{n} \\rfloor = 2014$\n\nRemember to put y... | 1958 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3 \\times 3 \\times 3$ cube such tha... | 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\nThe equation $166 \\times 56 = 8590$ is valid in some base $b \\ge 10$ (where $1, 6, 5, 8, 9, 0$ are digits in base $b$). Find... | 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 $Answer is the answer to the problem.\n\nAli defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that ea... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colo... | 53 |
[
{
"content": "Solve the following math problem step by 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 $x=\\sqrt{37-20\\sqrt{3}}$. Find the value of \\(\\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}\\).\n\nRemember to put your answe... | 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 $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance,... | 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\nCircles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not int... | 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 $Answer is the answer to the problem.\n\nJoel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strate... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a $5 \\times 5$ grid, we randomly place two \\emph{cars}, each occupying a single cell and randomly facing one of the four ... | 1148 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\\overline{AB}$ ... | 180 |
[
{
"content": "Solve the following math problem step by 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 non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$. Find the maximum value of the fol... | 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\nHow many integer pairs $(a, b)$ with $1 < a, b \\le 2015$ are there such that $\\log_a b$ is an integer?\n\nRemember to put yo... | 2082 |
[
{
"content": "Solve the following math problem step by 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 T-tetromino is formed by adjoining three unit squares to form a $1 \\times 3$ rectangle, and adjoining on top of the middle ... | 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 what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?\n\nRemember t... | 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 $Answer is the answer to the problem.\n\nThe sum of the following seven numbers is exactly 19: \n\\[a_1=2.56,\\ a_2=2.61,\\ a_3=2.65,\\ a_4=2.71,\\ a_5=2.79,\\ a_6=2.... | 61 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle. Let the $A$-altitude meet $BC$ at $D$, the $B$-altitude meet $AC$ at $E$, and let $T \\neq A$ be the ... | 112 |
[
{
"content": "Solve the following math problem step by 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 positive integers $n$ such that $\\sqrt{n} + \\sqrt{n + 1} < 11$.\n\nRemember to put your answer on its own... | 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 coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10... | 100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th,... | 240 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCount the sum of all four-digit positive integers that contain only odd digits in their decimal representation. Note that the ... | 3471875 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the se... | 260 |
[
{
"content": "Solve the following math problem step by 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 $f : \\mathbb{N} \\to \\mathbb{R}$ be a function, satisfying the following condition: for every integer $n > 1$, there exi... | 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 integer $n$ has exactly six positive divisors, which are $1 < a < b < c < d < n$. Let $k = a - 1$. If the $k$-th divisor (... | 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\nIn triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflec... | 288 |
[
{
"content": "Solve the following math problem step by 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 an isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircl... | 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\nCompute $\\left|\\sum_{i=1}^{2022} \\sum_{j=1}^{2022} \\cos\\left(\\frac{ij\\pi}{2023}\\right)\\right|.$\n\nRemember to put yo... | 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 $a_n$ be the least positive integer such that the sum of its digits is $n$. Find $a_1 + a_2 + a_3 + \\dots + a_{20}$.\n\nR... | 1074 |
[
{
"content": "Solve the following math problem step by 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 $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Frid... | 23 |
[
{
"content": "Solve the following math problem step by 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 $n$ is an integer $\\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number ... | 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\nLet $L, E, T, M, \\text{ and } O$ be digits that satisfy the equation $LEET + LMT = TOOL.$ Given that $O$ has the value of $0$... | 1880 |
[
{
"content": "Solve the following math problem step by 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 circumradius $17$, inradius $4$, circumcircle $\\Gamma$, and $A$-excircle $\\Omega$. Suppose the ... | 128 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the positive integer $n$ such that $32$ is the product of the real number solutions of the equation $x^{\\log_2(x^3)-n} =... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many positive integers $n \\leq 100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ ha... | 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\nWhat are the last $3$ digits of $1! + 2! + \\cdots + 100!$?\n\nRemember to put your answer on its own line after \"Answer:\"."... | 313 |
[
{
"content": "Solve the following math problem step by 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 $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. L... | 2102 |
[
{
"content": "Solve the following math problem step by 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 prime numbers $p$ such that there exist positive integers $q$ and $r$ satisfying the conditions: $p \\nmid q$, $3 \\n... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum.... | 511 |
[
{
"content": "Solve the following math problem step by 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 $m = 2007^{2008}$. How many natural numbers $n$ are there such that $n < m$ and $n(2n + 1)(5n + 2)$ is divisible by $m$ (i... | 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\nA tetrahedron $ABCD$ satisfies the following conditions: the edges $AB$, $AC$, and $AD$ are pairwise orthogonal, $AB=3$, and $... | 1998 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m) = 20$ and $s(33m) = 120$, what is the val... | 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 $Answer is the answer to the problem.\n\nA grasshopper jumps on the plane from an integer point (point with integer coordinates) to another integer point according to ... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when \\((1^2+1)(2^2+1)(3^2+1)\\dots(42^2+1)\\) is divided by \\(43\\). Your answer should be an integer bet... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many numbers between $1$ and $1,000,000$ are perfect squares but not perfect cubes?\n\nRemember to put your answer on its ... | 990 |
[
{
"content": "Solve the following math problem step by 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 gallons of a solution which is $15\\%$ alcohol do we have to mix with a solution that is $35\\%$ alcohol to make $250... | 175 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the following expression: \n\\[ \\sum_{i=0}^{\\frac{q-1}{2}}\\left\\lfloor\\frac{ip}{q}\\right\\rfloor + \\sum_{j=0}^... | 1001 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $3 \\times 6$ grid is filled with the numbers in the list $\\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\\}$ according to the follo... | 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\nDefine a sequence $\\{a_{i}\\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \\geq 1.$ Which integers between $0$ and $99$ incl... | 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\nMarie is painting a $4 \\times 4$ grid of identical square windows. Initially, they are all orange, but she wants to paint $4$... | 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\nConsider the $2015$ integers $n$, from $1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3... | 403 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Compute the largest positive integer $n$ such th... | 397 |
[
{
"content": "Solve the following math problem step by 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 $f(x) = 2^x + 3^x$. For how many integers $1 \\leq n \\leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \\dots,... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $16^4 + 16^2 + 1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.\n\nRemember to... | 264 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ ... | 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 $Answer is the answer to the problem.\n\nEvaluate the integer part of the number\n\\[ H = \\sqrt{1 + 2011^2 + \\frac{2011^2}{2012^2}} + \\frac{2011}{2012}. \\]\n\nReme... | 2012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a seq... | 500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all prime numbers \\( p \\) such that \\( p^2 \\) can be written as the sum of cubes of two positive integers, i.e., \\( ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA subset $S$ of the set of integers from 0 to 99 is said to have property $A$ if it is impossible to fill a crossword puzzle w... | 25 |
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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 $1, 7, 19, \\ldots$ be the sequence of numbers such that for all integers $n \\ge 1$, the average of the first $n$ terms i... | 261 |
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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 center of the circle passing through the midpoints of the sides of an isosceles triangle $ABC$ lies on the circumcircle of... | 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 $Answer is the answer to the problem.\n\nA number is called *trilegal* if its digits belong to the set \\(\\{1, 2, 3\\}\\) and if it is divisible by \\(99\\). How many... | 2025 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet there be $A = 1^{a_1} \\cdot 2^{a_2} \\cdot \\ldots \\cdot 100^{a_{100}}$ and $B = 1^{b_1} \\cdot 2^{b_2} \\cdot \\ldots \... | 0 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWork in base 3, where each digit can be 0, 1, or 2.\n\nA \"good\" number of size $n$ is defined as a number in which there are... | 8119 |
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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 the equations $x^2 + y^2 = 16$ and $xy = \\frac{9}{2}$. Find the sum, over all ordered pairs $(x, y)$ satisfying thes... | 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\nIf you roll six fair dice, let $p$ be the probability that exactly five different numbers appear on the upper faces of the six... | 133 |
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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\nHolding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point ... | 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 $Answer is the answer to the problem.\n\nFind the number of different quadruples \\((a, b, c, d)\\) of positive integers such that \\(ab = cd = a + b + c + d - 3\\).\n... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient 1, meets the x-axis at t... | 2023 |
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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 product of all possible real values for $k$ such that the system of equations\n\\[\nx^2 + y^2 = 80\n\\]\n\\[\nx^2 + y... | 960 |
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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 sequence of positive integers with $n$ terms satisfies $\\sum_{i=1}^{n} a_i = 2007$. Find the least positive integer $n$ suc... | 1018 |
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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 the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find... | 2491 |
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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\nAn alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as $U$, $W$, $XW$, $ZZ$, $V$, $Y$,... | 5390 |
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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 real numbers satisfying $2(\\sin a + \\cos a) \\sin b = 3 - \\cos b$. Find $3 \\tan^2 a + 4 \\tan^2 b$.\n\n... | 35 |
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