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\nLet $E$ and $F$ be the respective midpoints of $BC$ and $CD$ in a convex quadrilateral $ABCD$. Segments $AE$, $AF$, and $EF$ d... | 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 $ABCDA'B'C'D'$ be a rectangular parallelepiped, where $ABCD$ is the lower face and $A, B, C$, and $D$ are below $A', B', C... | 2015 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a m... | 2002 |
[
{
"content": "Solve the following math problem step by 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 reals $x \\ge 3$, let $f(x)$ denote the function \\[ f(x) = \\frac{-x + x\\sqrt{4x-3}}{2}. \\]\nLet $a_1, a_2, \\ldots$, b... | 4025 |
[
{
"content": "Solve the following math problem step by 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 the prime numbers less than $100$ which are one more than a multiple of 6.\n\nRemember to put your answer ... | 527 |
[
{
"content": "Solve the following math problem step by 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 ways a series of $+$ and $-$ signs can be inserted between the numbers $0, 1, 2, \\ldots, 12$ such that the... | 816 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFive men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is... | 59 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer is said to be a \"palindrome\" if it reads the same from left to right as from right to left. For example, ... | 495000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution ... | 4028 |
[
{
"content": "Solve the following math problem step by 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 $AC = 28$, $BC = 33$, and $\\angle ABC = 2\\angle ACB$. Compute the length of side $AB$.\n\nRemem... | 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\nA set $S$ consisting of $2019$ different positive integers has the following property: the product of every $100$ elements of ... | 1820 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircle $\\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA = 13$. The two tangents... | 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\nTriangle $ABC$ has a right angle at $A$, $AB=20$, and $AC=21$. Circles $\\omega_A$, $\\omega_B$, and $\\omega_C$ are centered ... | 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\nCompute the least possible value of $ABCD - AB \\times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are ... | 109 |
[
{
"content": "Solve the following math problem step by 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 area enclosed by the graph of $|x| + |2y| = 12$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 144 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest integer $N$ for which there exists a $6 \\times N$ table $T$ that has the following properties:\n\n- Eve... | 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\nTwo boys are given a bag of potatoes, each bag containing $150$ tubers. They take turns transferring the potatoes, where in ea... | 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 least positive integer $n$ such that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily ... | 135 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle $ABCD$ is divided into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints... | 150 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p(n)$ denote the product of the decimal digits of a positive integer $n$. Compute the sum $p(1) + p(2) + \\ldots + p(2001... | 184320 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that a polynomial of the form $p(x) = x^{2010} \\pm x^{2009} \\pm \\cdots \\pm x \\pm 1$ has no real roots. What is th... | 1005 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTo celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits... | 1092 |
[
{
"content": "Solve the following math problem step by 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 $f(x) = -\\frac{2x}{4x+3}$ and $g(x) = \\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x) = g(f(h^n(x)))$, where $h^1(x) = ... | 128 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $n$ such that $$ \\frac{n+11}{\\sqrt{n-1}} $$ is an integer.\n\nRemember to put your ans... | 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\nSuppose $x, y, z$, and $w$ are positive reals such that:\n\\[ x^2 + y^2 - \\frac{xy}{2} = w^2 + z^2 + \\frac{wz}{2} = 36 \\]\n... | 960 |
[
{
"content": "Solve the following math problem step by 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 $x$ is a real number satisfying the equation: \n\\[ x^2 - 990x + 1 = (x + 1)\\sqrt{x} \\] \nFind the value of \\( \\... | 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\nA picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border forms a rectangl... | 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 six sides of a convex hexagon $A_1 A_2 A_3 A_4 A_5 A_6$ are colored red. Each of the diagonals of the hexagon is colored e... | 94 |
[
{
"content": "Solve the following math problem step by 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 three-digit number $x$ in base 10 has a units-digit of 6. When $x$ is written in base 9, the second digit of the number is 4... | 446 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest positive integer \\( n \\) satisfying the following inequality:\n\\[ n^{2006} < 7^{2007} \\]\n\nRemember t... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value o... | 2350 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the sum: \n\\[\n\\frac{3!+4!}{2(1!+2!)} + \\frac{4!+5!}{3(2!+3!)} + \\cdots + \\frac{12!+13!}{11(10!+11!)}\n\\]\n\nRe... | 95 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\Gamma_1$ and $\\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that t... | 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\nFor integers $a$, $b$, $c$, and $d$, the polynomial $p(x) = ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the mi... | 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 $Answer is the answer to the problem.\n\nThere are 15 people at a party; each person has 10 friends. To greet each other, each person hugs all their friends. How many ... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer $n$ is considered \\textit{magical} if the following condition holds: \n\\[ \\lfloor \\sqrt{\\lceil \\sqrt{... | 1330 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the country of Óxisz, the minister of finance observed at the end of the tax census that the sum of properties of any two n... | 1000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the order $2,4,\... | 55 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials: \n\\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \\] \nand \n\\... | 2400 |
[
{
"content": "Solve the following math problem step by 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 positive prime $p$ which satisfies the congruence $p + p^{-1} \\equiv 25 \\pmod{143}$. Here, $p^{-1}$ d... | 269 |
[
{
"content": "Solve the following math problem step by 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 $\\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\\overline{AB}$ and $\\overline{A... | 878 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the ten numbers $ar, ar^2, ar^3, \\ldots, ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, determi... | 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 $Answer is the answer to the problem.\n\nEach time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start ... | 142 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x)$. There are relatively prime positive integers such that $x = -... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral with $AC \\perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the d... | 231 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Ale... | 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 $P$ be the portion of the graph of\n\\[ y=\\frac{6x+1}{32x+8} - \\frac{2x-1}{32x-8} \\]\nlocated in the first quadrant (no... | 433 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(x) = x + 1$ and $Q(x) = x^2 + 1$. Consider all sequences $\\langle (x_k, y_k) \\rangle_{k \\in \\mathbb{N}}$ such that ... | 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\nFind the sum of all positive integers $n$ such that:\n\n$$ \\frac{\\sqrt{n}}{2} + \\frac{30}{\\sqrt{n}} $$\n\nis an integer.\n... | 1040 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAnna and Berta play a game in which they take turns removing marbles from a table. Anna takes the first turn. At the beginning... | 131070 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $T$ be a $30$-$60$-$90$ triangle with a hypotenuse of length $20$. Three circles, each externally tangent to the other two... | 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\nA line with a negative slope passing through the point $(18,8)$ intersects the $x$-axis at $(a,0)$ and the $y$-axis at $(0,b)$... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a trapezoid with $AB \\parallel CD$, $AB$ has length $1$, and $CD$ has length $41$. Let points $X$ and $Y$ lie o... | 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\nBrian starts at the point $(1,0)$ in the plane. Every second, he performs one of two moves: he can move from $(a,b)$ to $(a-b,... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( f \\) be a function from \\( \\{1, 2, \\ldots, 22\\} \\) to the positive integers such that \\( mn \\mid f(m) + f(n) \... | 2016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGus needs to create a list of $250$ positive integers, where each integer is equal to the number of other numbers in the list ... | 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\nLet $f$ be a function from positive integers to positive integers where $f(n) = \\frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$... | 496 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA palindrome is a word that reads the same backward as forward. Examples include OMO, lepel, and parterretrap. Determine how m... | 640 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven $\\triangle ABC$ with $\\angle A = 15^{\\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $B... | 256 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each w... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,... | 503 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\\triangle{ABC}$ is similar to $\\triangle{ACD}$. ... | 169 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBen starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this ... | 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 $Answer is the answer to the problem.\n\nOf all positive integral solutions \\((x,y,z)\\) to the equation \n\\[x^3+y^3+z^3-3xyz=607,\\] \ncompute the minimum possible ... | 1213 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAdd one pair of brackets to the expression $1+2\\times 3+4\\times 5+6$ so that the resulting expression has a valid mathematic... | 77 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe diagram shows a semicircle with a diameter of $20$ and the largest circle that fits inside the semicircle. The area of the... | 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\nThe sides of a triangle are $x$, $2x+1$, and $x+2$, where $x$ is a positive rational number. One of the angles of the triangle... | 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 $Answer is the answer to the problem.\n\nThe squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.\n\nRemember to put you... | 1008 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle $ABC$ is equilateral. Points $D$ and $E$ are the midpoints of segments $BC$ and $AC$, respectively. $F$ is the point ... | 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\nQuadrilateral $ABCD$ satisfies $AB = 8$, $BC = 5$, $CD = 17$, and $DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Sup... | 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 $n$ be an integer. We consider $s(n)$, the sum of the 2001 powers of $n$ with the exponents from 0 to 2000. Thus, $s(n) = ... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ in... | 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\nTriangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and... | 92 |
[
{
"content": "Solve the following math problem step by 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 the product of its decimal digits equals $n^2 - 10n - 22$.\n\nRemember to put your answ... | 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 $n \\ge 3$. Suppose $a_1, a_2, \\ldots, a_n$ are $n$ distinct pairwise real numbers. In terms of $n$, find the smallest po... | 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 $Answer is the answer to the problem.\n\nFind the number of triples of real numbers \\((a, b, c)\\) such that \\(a^{20} + b^{20} + c^{20} = a^{24} + b^{24} + c^{24} = ... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. ... | 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\nFor each of the $9$ positive integers $n, 2n, 3n, \\ldots, 9n$, Alice takes the first decimal digit (from the left) and writes... | 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 $P(x) = x^3 + ax^2 + bx + c$ be a polynomial where $a, b, c$ are integers and $c$ is odd. Let $p_i$ be the value of $P(x)$... | 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\nLet $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \\[ zP(y) = P(y - n) + P(y + n) \\] for ... | -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\nLet $a, b, c$ be integers, not all the same, with $a, b, c \\geq 4$ that satisfy the equation:\n\\[ 4abc = (a + 3)(b + 3)(c + ... | 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 $Answer is the answer to the problem.\n\nThe battery life on a computer decreases at a rate proportional to the display brightness. Austin starts his day with both his... | 125 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size 25. For any finite subset of... | 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 $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real... | 1266 |
[
{
"content": "Solve the following math problem step by 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 table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $... | 2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$. Define $\\{x \\} = x - \\lfloor x \\rfloor$. Fo... | 145 |
[
{
"content": "Solve the following math problem step by 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 $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \n\... | 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\nFor how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13$?\n\... | 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\nIf $a$ and $b$ are nonzero real numbers such that $|a| \\neq |b|$, compute the value of the expression\n\\[\n \\left( \\frac{... | -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\nIn how many ways can one choose distinct numbers $a$ and $b$ from the set \\( \\{1, 2, 3, \\ldots, 2005\\} \\) such that \\( a... | 401802 |
[
{
"content": "Solve the following math problem step by 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 $8 \\times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \\times 1$ cells in the u... | 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\nIn a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their di... | 360 |
[
{
"content": "Solve the following math problem step by 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 polynomial of degree at most 6 that satisfies $P(k) = k!$ for $k = 0, 1, 2, 3, 4, 5, 6$. Compute the value of ... | 3186 |
[
{
"content": "Solve the following math problem step by 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 $3 \\times 3 \\times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways of select... | 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\nThree parallel lines $L_1$, $L_2$, and $L_3$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSeven boys and three girls are playing basketball. In how many different ways can they make two teams of five players each, en... | 105 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of pairs of positive integers $(m, n)$, with $m \\leq n$, such that the least common multiple (LCM) of $m$ and... | 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\nFor each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into t... | 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 $Answer is the answer to the problem.\n\n$f$ is a function defined on integers and satisfies $f(x) + f(x+3) = x^2$ for every integer $x$. If $f(19) = 94$, then calcula... | 4335 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.