prompt listlengths 1 1 | label stringlengths 1 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\nHow many subsets of the set $\\{1, 2, \\ldots, 11\\}$ have median 6?\n\nRemember to put your answer on its own line after \"An... | 351 |
[
{
"content": "Solve the following math problem step by 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 ant is walking on the edges of an icosahedron with a side length of $1$. Compute the length of the longest path that the an... | 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\nEach unit square of a $4 \\times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid... | 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 sum of all distinct positive divisors of the number $104060401$.\n\nRemember to put your answer on its own line after... | 105101005 |
[
{
"content": "Solve the following math problem step by 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 class with $23$ students, each pair of students has watched a movie together. Let the set of movies watched by a student ... | 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\nFind $x$ such that:\n\\[\n\\frac{\\frac{5}{x-50} + \\frac{7}{x+25}}{\\frac{2}{x-50} - \\frac{3}{x+25}} = 17.\n\\]\n\nRemember ... | 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 $Answer is the answer to the problem.\n\n$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of... | 38 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \\ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of... | 763 |
[
{
"content": "Solve the following math problem step by 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 four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal poss... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest possible value of $|12^m - 5^n|$, where $m$ and $n$ are positive integers?\n\nRemember to put your answer... | 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\nIn $\\triangle ABC$, $\\angle ABC = 75^\\circ$ and $\\angle BAC$ is obtuse. Points $D$ and $E$ are on $AC$ and $BC$, respectiv... | 85 |
[
{
"content": "Solve the following math problem step by 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 $\\oplus$ denote the xor binary operation. Define $x \\star y = (x+y) - (x \\oplus y).$ Compute \n\\[ \\sum_{k=1}^{63} (k ... | 2880 |
[
{
"content": "Solve the following math problem step by 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 the set of positive integers that are the product of two consecutive integers. Let $B$ be the set of positive integ... | 216 |
[
{
"content": "Solve the following math problem step by 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 complex quartic polynomial $Q$ is called quirky if it has four distinct roots, one of which is the sum of the other three. T... | 720 |
[
{
"content": "Solve the following math problem step by 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 $\\omega$ be a circle. Let $E$ be on $\\omega$ and $S$ outside $\\omega$ such that line segment $SE$ is tangent to $\\omeg... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the fourteen numbers, $1^4, 2^4, \\ldots, 14^4$. Find the smallest natural number $n$ such that these numbers leave d... | 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\nSuppose that $N$ is a three-digit number divisible by $7$ such that upon removing its middle digit, the remaining two-digit nu... | 154 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoint $P_1$ is located 600 miles west of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives east at a speed of 50 mp... | 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\nLet $S$ be a set of sequences of length 15 formed by using the letters $a$ and $b$, such that every pair of sequences in $S$ d... | 2048 |
[
{
"content": "Solve the following math problem step by 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 nonnegative solutions to the equation $2x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} = 3$ is ___... | 174 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\\overline{CD}$, and let $F$ be the point on segment $... | 46 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn squared paper, a square is delimited, which comprises $25$ unit squares. How many rectangles are there in this square that ... | 225 |
[
{
"content": "Solve the following math problem step by 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 smallest positive integer such that $m^2 + (m+1)^2 + \\cdots + (m+10)^2$ is the square of a positive integer $n... | 95 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many unordered triples $A, B, C$ of distinct lattice points in $0 \\leq x, y \\leq 4$ have the property that $2[ABC]$ is a... | 300 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider real numbers $A$, $B$, \\dots, $Z$ such that:\n\\[ \n EVIL = \\frac{5}{31}, \\; \n LOVE = \\frac{6}{29}, \\text{ an... | 579 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA finite non-empty set of integers is called $3$-good if the sum of its elements is divisible by $3$. Find the number of $3$-g... | 351 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ is divisible by $3$ and $5$, but not by $2$. If $n > 20$, what is the smallest possible value of $n$?\n... | 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 $Answer is the answer to the problem.\n\nGiven that \\(x, y,\\) and \\(z\\) are real numbers satisfying \\(x + y + z = 10\\) and \\(x^2 + y^2 + z^2 = 50\\), find the m... | 1250 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 47 students in a classroom with seats arranged in 6 rows $\\times$ 8 columns, and the seat in the $i$-th row and $j$... | 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\nFind the number of subsets $A \\subset M = \\{2^0, 2^1, 2^2, \\dots, 2^{2005}\\}$ such that the equation $x^2 - S(A)x + S(B) =... | 1003 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf\n\\[\n\\sum_{n=1}^{\\infty}\\frac{\\frac{1}{1} + \\frac{1}{2} + \\dots + \\frac{1}{n}}{\\binom{n+100}{100}} = \\frac{p}{q}\... | 9901 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c$ be the roots of the polynomial $x^3 - 9x^2 + 11x - 1 = 0$. Define $s = \\sqrt{a} + \\sqrt{b} + \\sqrt{c}$. Find ... | -37 |
[
{
"content": "Solve the following math problem step by 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 4-digit numbers have exactly $9$ divisors from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$?\n\nRemember to put your answer o... | 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 a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the tri... | 74 |
[
{
"content": "Solve the following math problem step by 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 natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers.\n\nRemembe... | 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\nIn a triangle $ABC$, the median $AD$ divides $\\angle BAC$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicu... | 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\nFind the number of ordered pairs $(a, b)$ of positive integers such that $a < b$ and $100 \\leq a, b \\leq 1000$ satisfying $\... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTowers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as ... | 1033 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor some complex number $\\omega$ with $|\\omega| = 2016$, there is some real $\\lambda>1$ such that $\\omega$, $\\omega^{2}$,... | 4032 |
[
{
"content": "Solve the following math problem step by 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 the decimal digits of the number $(10^{2021} + 2021)^2$?\n\nRemember to put your answer on its own line after ... | 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\nJacob likes to watch Mickey Mouse Clubhouse! One day, he decides to create his own Mickey Mouse head with two circles $\\omega... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\\triangle ABC$ ... | 325 |
[
{
"content": "Solve the following math problem step by 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 figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same poin... | 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 $Answer is the answer to the problem.\n\nCompute the sum of the roots of the equation \\((2 - x)^{2005} + x^{2005} = 0\\).\n\nRemember to put your answer on its own li... | 2004 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAmong $100$ points in the plane, no three are collinear, exactly $4026$ pairs are connected by line segments. Each point is th... | 1037 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe triangle $ABC$ has side lengths $AB = 39$, $BC = 57$, and $CA = 70$. The median $\\overline{AD}$ is divided into three con... | 21 |
[
{
"content": "Solve the following math problem step by 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 maximal number of checkers that can be placed on an $8 \\times 8$ checkerboard so that each checker stands on the ... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest integer $n > 1$ such that $n^2(n - 1)$ is divisible by 2009.\n\nRemember to put your answer on its own ... | 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\nA square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoint... | 1 |
[
{
"content": "Solve the following math problem step by 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 integer not exceeding the sum $\\sum_{n=1}^{1599} \\frac{1}{\\sqrt{n}}$?\n\nRemember to put your answer o... | 78 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) has an area of \\( 240 \\). Points \\( X, Y, Z \\) lie on sides \\( AB, BC, \\) and \\( CA, \\) respect... | 122 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?\n\nRemember to put your answer on its own line... | 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 distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C$ in deg... | 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 $Answer is the answer to the problem.\n\nConsider all 6-digit numbers of the form \\( abccba \\) where \\( b \\) is odd. Determine the number of all such 6-digit numbe... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that three roots of \\( f(x) = x^4 + ax^2 + bx + c \\) are \\( 2 \\), \\( -3 \\), and \\( 5 \\), what is the value of \\... | 79 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe points $A$, $B$, $C$, $D$, $E$ lie on a line $\\ell$ in this order. Suppose $T$ is a point not on $\\ell$ such that $\\ang... | 954 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEighteen people are standing in a socially-distanced line to enter a grocery store. Five people are wearing a black mask, $6$ ... | 116 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a driver's seat, a passenger... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Jones family has five children, and the Smith family has three children. Among these eight children, there are five girls ... | 67 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle $ABC$ with $AC = 20$ is inscribed in a circle $\\omega$. A tangent $t$ to $\\omega$ is drawn through $B$. The dista... | 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\nLet $A$, $B$, $C$, $D$, $E$, and $F$ be 6 points around a circle, listed in clockwise order. We have $AB = 3\\sqrt{2}$, $BC = ... | 225 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is an interval $[a, b]$ that is the solution to the inequality $|3x - 80| \\le |2x - 105|$. Find $a + b$.\n\nRemember to... | 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 $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:\n- $P$ passes through $A$ and $B$,... | 2402 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider \\(13\\) marbles that are labeled with positive integers such that the product of all \\(13\\) integers is \\(360\\).... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a square with $AB = 6$. A point $P$ inside the square is 2 units away from side $BC$ and 3 units away from side ... | 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\nFor each set of five integers $S= \\{a_1, a_2, a_3, a_4, a_5\\}$, let $P_S$ be the product of all differences between two of t... | 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 $Answer is the answer to the problem.\n\nJohn and Mary each select a natural number and tell them to Bill. Bill writes down their sum and product on two separate piece... | 1001 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the sum: \\( \\frac{1}{a_1} + \\frac{1}{a_2} + \\ldots + \\frac{1}{a_{20}} \\), where \\( a_n = \\sqrt{1 + \\left(1 -... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered triplets $(a, b, c)$ of positive integers satisfy the inequality $30a + 50b + 70c \\leq 343$?\n\nRemember to ... | 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 $Answer is the answer to the problem.\n\nFive teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every ot... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be a subset of $\\{1, 2, 3, \\cdots, 1989 \\}$ in which no two members differ by exactly $4$ or by exactly $7$. What i... | 905 |
[
{
"content": "Solve the following math problem step by 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 quadrilateral with $\\angle ADC = 70^{\\circ}$, $\\angle ACD = 70^{\\circ}$, $\\angle ACB = 10^{\\circ}$, and ... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $70$ foot pole stands vertically in a plane and is supported by three $490$ foot wires. All wires are attached to the top of... | 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\nCompute the remainder when $1^{2547} + 2^{2547} + \\ldots + 2547^{2547}$ is divided by $25$.\n\nRemember to put your answer on... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x, y$ be distinct positive real numbers satisfying:\n\\[\n\\frac{1}{\\sqrt{x + y} - \\sqrt{x - y}} + \\frac{1}{\\sqrt{x +... | 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\nFind the smallest positive three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$.\n\nRemember to put your answer o... | 102 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle with radius $r$ has an area of $505$. Compute the area of a circle with a diameter of $2r$.\n\nRemember to put your a... | 2020 |
[
{
"content": "Solve the following math problem step by 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 triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ su... | 623 |
[
{
"content": "Solve the following math problem step by 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 all pairs of points $(a, b, c)$ and $(d, e, f)$ in the $3$-D coordinate system such that $ad + be + cf = -2023$. What... | 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\nFind the number of pairs of integers $(x, y)$ such that $x^2 + 2y^2 < 25$.\n\nRemember to put your answer on its own line afte... | 55 |
[
{
"content": "Solve the following math problem step by 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 real numbers $B$, $M$, and $T$, we have the following conditions: \n1. $B^2 + M^2 + T^2 = 2022$ \n2. $B + M + T = 72$\n\... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKara rolls a six-sided die six times, and notices that the results satisfy the following conditions:\n- She rolled a $6$ exact... | 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 $Answer is the answer to the problem.\n\n$N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisibl... | 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\nFor each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n! + 1$ and $(n+1)!$. For $n < 100... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest positive integer $n$ for which there exist pairwise different sets $\\mathbb{S}_1 , \\ldots, \\mathbb{S}... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPrincess Pear has $100$ jesters with heights $1, 2, \\dots, 100$ inches. On day $n$ with $1 \\leq n \\leq 100$, Princess Pear ... | 384160000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCalculate $\\displaystyle{\\sum_{n=1}^\\infty\\left(\\lfloor\\sqrt[n]{2010}\\rfloor-1\\right)}$ where $\\lfloor x\\rfloor$ is ... | 2077 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than 2005 are relatively prime to 1001?\n\nRemember to put your answer on its own line after \... | 1442 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) \\cdots (1 - 2008x)$.\n\nRemember to put your... | -1004 |
[
{
"content": "Solve the following math problem step by 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 set of monic polynomials with integer coefficients of the least degree, with root $k \\cdot \\cos\\lef... | 167 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\text{HOW}$, $\\text{BOW}$, and $\\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given th... | 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\nFind the least positive integer $N$ such that the only values of $n$ for which $1 + N \\cdot 2^n$ is prime are multiples of $1... | 556 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $1 \\times 5$ rectangle is divided into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at ... | 364 |
[
{
"content": "Solve the following math problem step by 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 number of $1$'s in the binary representation of $n$. Compute \\[ \\frac{1}{255}\\sum_{0 \\leq n < 16} 2^... | 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 $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ an... | 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 each positive integer $k$, denote $C(k)$ as the sum of its distinct prime divisors. For example, $C(1) = 0$, $C(2) = 2$, a... | 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 $Answer is the answer to the problem.\n\n是否存在一个无穷正整数数列，使得对任意正整数 k，数列的任意连续 k 项之和均为 k+1 的倍数？\n让我们一步一步地思考。请以“Answer: \\boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJaviera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the b... | 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\nFind all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square.\n\nRemember to put your answer on its own line after \"... | 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\nIn a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th swi... | 1024 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.