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\nCompute $ax^{2018} + by^{2018}$, given that there exist real numbers $a$, $b$, $x$, and $y$ which satisfy the following four e... | -417 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.\n\nRemember ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that the base-17 integer $\\overline{8323a02421_{17}}$ (where $a$ is a base-17 digit) is divisible by $16_{10}$, find $a... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\Gamma_1$, $\\Gamma_2$ be two circles, where $\\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. P... | 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\nTriangle $ABC$ has $\\overline{AB} = 5$, $\\overline{BC} = 4$, $\\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and ... | 255 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGreta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve p... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the sum:\n\\[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \\dots - 2010 \\]\nwhere every three consecutive \\( + \\) ... | 401799 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.\n\nRemember to put your... | 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\nThe number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?\n\nRemembe... | 3004 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiraldo wrote five distinct natural numbers on the vertices of a pentagon. Next, he wrote on each side of the pentagon the lea... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPaula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a tria... | 109 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves fr... | 126 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRight triangle $\\triangle ABC$ with a right angle at $A$ has $AB = 20$ and $AC = 15$. Point $D$ is on $AB$ with $BD = 2$. Poi... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of ordered quintuples of nonnegative integers \\((a_1, a_2, a_3, a_4, a_5)\\) such that \\(0 \\leq a_1, a_2... | 6528 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that ... | 160402 |
[
{
"content": "Solve the following math problem step by 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 unit square. If $\\angle PAC = \\angle PCD$, find the length $BP$.\n\nRemember to put your answer on its own line ... | 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 $Answer is the answer to the problem.\n\nThe Tournament of Towns is held once per year. This time, the year of its autumn round is divisible by the number of the tourn... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIt is known that $2018(2019^{39} + 2019^{37} + \\cdots + 2019) + 1$ is prime. How many positive factors does $2019^{41} + 1$ h... | 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 $S$ be the sum of all $x$ such that $1 \\leq x \\leq 99$ and \\[ \\{x^2\\} = \\{x\\}^2. \\] Compute $\\lfloor S \\rfloor$.... | 641999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of positive integers $(a,b)$ such that $a+b$ is a prime number, $1 \\leq a, b \\leq 100$, and... | 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\nUnconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$, with $1$ and $2$ appea... | 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\nFor a positive integer $n$, define the $n$th triangular number $T_n$ to be $\\frac{n(n+1)}{2}$, and define the $n$th square nu... | 1954 |
[
{
"content": "Solve the following math problem step by 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 called a *primary divisor* if for every positive divisor $d$ of $n$, at least one of the numbers $d ... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider an odd prime $p$. For each $i$ in the set \\( \\{1, 2, \\ldots, p-1\\} \\), let \\( r_i \\) be the remainder of \\( i... | 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\nConsider a board with $n$ rows and $4$ columns. In the first line, there are $4$ zeros (one in each column). Each subsequent l... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_1 = \\frac{1}{20}$, $x_2 = \\frac{1}{13}$, and \\[ x_{n+2} = \\frac{2x_nx_{n+1}(x_n + x_{n+1})}{x_n^2 + x_{n+1}^2} \\] ... | 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 $Q$ be a polynomial \\[Q(x)=a_0+a_1x+\\cdots+a_nx^n,\\] where $a_0,\\ldots,a_n$ are nonnegative integers. Given that $Q(1)... | 254 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the diagram below, $ABCDE$ is a regular pentagon. The line $\\overline{AG}$ is perpendicular to $\\overline{CD}$, and $\\ov... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to arrange the numbers $1$ through $8$ into a $2 \\times 4$ grid such that:\n1. The sum of the numbers... | 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\nLet $f : \\mathbb{N} \\to \\mathbb{N}$ be a function such that $f(f(1995)) = 95$, $f(xy) = f(x)f(y)$, and $f(x) \\le x$ for al... | 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\nSitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \\le 10^{10}$. Interestingly, Celia, si... | 4999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsing the digits $1, 2, 3, 4, 5, 6$ without repetition, form three two-digit numbers. Calculate the number of different sums t... | 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 $f(n)$ be the number of times you have to hit the $\\sqrt{}$ key on a calculator to get a number less than $2$ starting fr... | 242 |
[
{
"content": "Solve the following math problem step by 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 following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_0, x_1, \\ldots$ be a sequence of real numbers such that \\(x_n = \\frac{1 + x_{n -1}}{x_{n - 2}}\\) for \\(n \\geq 2\\... | 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\nToday is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \\ldots... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \\[ P(42) = P(69) = P(96) = P(123) = 13. \\] C... | 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\nThe polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r$, $s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational a... | 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\nA substring of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different... | 576 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMr. Ambulando is at the intersection of $5^{\\text{th}}$ and $\\text{A St}$, and needs to walk to the intersection of $1^{\\te... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest positive integer $k$ such that $\\phi(\\sigma(2^k)) = 2^k$. Here, $\\phi(n)$ denotes the number of positive i... | 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\nThere exists a unique pair of polynomials $P(x)$ and $Q(x)$ such that:\n\\[ \\begin{align*} P(Q(x)) &= P(x)(x^2-6x+7) \\\\ Q(P... | -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\nLet $X_0$ be the interior of a triangle with side lengths $3$, $4$, and $5$. For all positive integers $n$, define $X_n$ to be... | 4112 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$, and $d$ be real numbers such that \\[ a^2 + 3b^2 + \\frac{c^2 + 3d^2}{2} = a + b + c + d - 1. \\] Find $100... | 527 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negativ... | 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 $Answer is the answer to the problem.\n\nThe license plate of each automobile in Iran consists of a two-digit and a three-digit number, as well as a letter from the Pe... | 30525 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $n$ for which $|2^n + 5^n - 65|$ is a perfect square.\n\nRemember to put your answer on ... | 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 $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \\cdo... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prim... | 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\nCircle $\\omega_1$ and $\\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and interse... | 303 |
[
{
"content": "Solve the following math problem step by 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 positive real numbers $a, b, c$ satisfy the following equations:\n\n$$\\frac{a}{2b+1} + \\frac{2b}{3c+1} + \\frac{3c}{a+1}... | 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 diagram below shows $\\triangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respec... | 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\nTriangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ a... | 208 |
[
{
"content": "Solve the following math problem step by 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 prime number $q$ is called a 'Kowai' number if $q = p^2 + 10$, where $q$, $p$, $p^2-2$, $p^2-8$, and $p^3+6$ are all prime n... | 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\n1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any indi... | 94605 |
[
{
"content": "Solve the following math problem step by 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 of the number $1 \\binom{2008}{0} + 2 \\binom{2008}{1} + \\ldots + 2009 \\binom{2008}{2008}$ when divide... | 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\nLet $LOVER$ be a convex pentagon such that $LOVE$ is a rectangle. Given that $OV = 20$ and $LO = VE = RE = RL = 23$, compute t... | 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\nConsider the $2 \\times 3$ rectangle below. We fill in the small squares with the numbers $1, 2, 3, 4, 5, 6$ (one per square).... | 157 |
[
{
"content": "Solve the following math problem step by 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 \\[ \\frac{1}{729} \\sum_{a=1}^{9} \\sum_{b=1}^9 \\sum_{c=1}^9 \\left( abc+ab+bc+ca+a+b+c \\right). \\]\n\nRemember t... | 215 |
[
{
"content": "Solve the following math problem step by 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 integral:\n\n$$ \\int_0^1 \\frac{6x + 5a\\sqrt{x}}{4\\sqrt{x + a\\sqrt{x}}}\\, dx $$\n\nwhere $a = 2007 \\times 2... | 2008 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the absolute value of the sum \\[ \\lfloor 2013\\sin{0^\\circ} \\rfloor + \\lfloor 2013\\sin{1^\\circ} \\rfloor + \\... | 178 |
[
{
"content": "Solve the following math problem step by 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 constant $k$ such that the sum of all $x \\ge 0$ satisfying $\\sqrt{x}(x+12)=17x-k$ is $256.$\n\nRemember to put your... | 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\nFind the distance $\\overline{CF}$ in the diagram below where $ABDE$ is a square, and angles and lengths are as given:\n\n\\[\... | 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 $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\\angle AOB$ is right. Points $C$ and $D$ lie on... | 44 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar ... | 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\nFind the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \\cdots, x_n$ of integers with the follo... | 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\nIf $a$, $b$ are integers and $s = a^3 + b^3 - 60ab(a + b) \\geq 2012$, find the least possible value of $s$.\n\nRemember to pu... | 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\nTwo circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line $AB$ is a common internal tangent, and ... | 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\nJerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he t... | 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\nConsider a $2008 \\times 2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard su... | 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\nCall a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the d... | 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\nThree numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}$. Thr... | 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\nEvaluate $ \\sqrt{\\pi \\int_0^{2008} x|\\sin \\pi x|\\, dx}$\n\nRemember to put your answer on its own line after \"Answer:\"... | 2008 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the expected sum of elements in a subset of $\\{1, 2, 3, \\ldots, 2020\\}$ (including the empty set) chosen uniformly ... | 1020605 |
[
{
"content": "Solve the following math problem step by 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 intersects an equilateral triangle $\\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$\\{10a+b \\mid 1 \\leq a \\leq 5, 1 ... | 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\nLet $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \\le i \\le 3$, let $D_i$, $E_i$, and $F... | 917 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a sequence defined as follows:\n- $a(2) = 0$\n- $a(3) = 1$\n- For $n \\geq 4$, $a(n) = a\\left(\\left\\lfloor \\frac{... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, \\ldots, x_{10})$ of nonnegative integers such that:\n\\[ x_1^2 + x... | 1006 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its are... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\( x \\) be the answer to this question. Find the value of \\( 2017 - 2016x \\).\n\nRemember to put your answer on its ow... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe value of the infinite series $$\\sum_{n=2}^\\infty \\dfrac{n^4+n^3+n^2-n+1}{n^6-1}$$ can be expressed as $\\frac{p}{q}$ wh... | 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\nA graph has $30$ points, and each point has $6$ edges. Find the total number of triples such that each pair of points is eithe... | 4060 |
[
{
"content": "Solve the following math problem step by 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 integer $n$ with the following property: for every triple of positive integers $x, y, z$, with... | 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 square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find... | 1168 |
[
{
"content": "Solve the following math problem step by 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 trapezoid $ABCD$, $AD$ is parallel to $BC$, with the side lengths $AB = 6$, $BC = 7$, $CD = 8$, and $AD = 17$. If the sides... | 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 odd three-digit positive integers $x$ satisfying $x^2 \\equiv 1 \\pmod{8}$ and the product of the digits of... | 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 $f : \\mathbb{R} \\to \\mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x +... | 167 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 100 houses on the street, divided into 50 pairs. Each pair consists of houses directly opposite each other across th... | 199 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou have a $3 \\times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical ... | 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\nLet $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees... | 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\nThere are five cities $A, B, C, D, E$ on a certain island. Each city is connected to every other city by road. In how many way... | 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\nTwo players play a game, starting with a pile of $N$ tokens. On each player's turn, they must remove $2^n$ tokens from the pil... | 1346 |
[
{
"content": "Solve the following math problem step by 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 $a_1 = 1$, and for all $n \\ge 2$, \\[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \\ldots + (n-1)a_1. \\] Furthermore,... | 2022 |
[
{
"content": "Solve the following math problem step by 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 $ABC$ have $\\angle A = 70^\\circ$, $\\angle B = 60^\\circ$, and $\\angle C = 50^\\circ$. Extend altitude $BH$ pa... | 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 $P$ be a parabola with focus $F$ and directrix $\\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $... | 140 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\\{1, 2, \\ldots, 2024\\}$,... | 1012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\\pm 1$. Imagine... | 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\nIn $\\triangle ABC$, let point $D$ be the foot of the altitude from $A$ to $BC$. Suppose that $\\angle A = 90^{\\circ}$, $AB -... | 150 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f : \\mathbb{N} \\to \\mathbb{N}$ be a function such that the following conditions hold:\n\n1. $f(1) = 1.$\n\n2. $\\frac{... | 1012 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.