prompt listlengths 1 1 | label stringlengths 1 210 |
|---|---|
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Ass... | 61 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 distinct similar rhombi share a diagonal. The smaller rhombus has an area of $1$, and the larger rhomb... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 rectangle $ABCD$ with center $O$, where $AB=10$ and $BC=8$, circle $\\gamma$ is centered at $O$ and is ... | 16808 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 prime numbers $p$ between $100$ and $200$ for which the congruence equation $x^{11} + y... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a circle, $2018$ points are marked. Each of these points is labeled with an integer. Let each number be... | 1008 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nStacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First, she goes to the lottery... | 1023 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ be a positive number. Find the number of integer solutions $x$ of the inequality $2 < xa < 3$, giv... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.\n\nRem... | 15 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is a square of checkered paper measuring $102 \\times 102$ squares and a connected figure of unknown... | 102 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 four-digit multiples of $8$ are greater than $2008$?\n\nRemember to put your answer on its own li... | 998 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of... | 706 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $(a_n)_{n \\geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n... | 17 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $a, b, c \\ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$, then what is the value of $... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last three digits in the product $1 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot \\ldots \\cdot 2009 \\cdot ... | 875 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $m$ such that $529^n + m \\cdot 132^n$ is divisible by $262417$ fo... | 1984 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWilliam writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the black... | 173 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $ \\#\\left\\{ (x,y)\\in\\mathbb{N}^2\\bigg| \\frac{1}{\\sqrt{x}} -\\frac{1}{\\sqrt{y}} =\\frac{1}{20... | 82 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 area of the region in the $xy$-plane satisfying the inequality \n\\[\\min_{1 \\le n \\le 10} \\max\\le... | 210 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 integers from $1$ through $9$ inclusive are placed in the squares of a $3 \\times 3$ grid. Each square... | 17 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $a, b$ are positive real numbers such that $a + a^2 = 1$ and $b^2 + b^4 = 1$. Compute $a^2 + b^2$.... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPrinceton’s Math Club recently bought a stock for $\\$2$ and sold it for $\\$9$ thirteen days later. Given... | 273 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are three bags:\n1. Bag A contains 3 green candies and 1 red candy.\n2. Bag B contains 2 green candi... | 217 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the star-shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the ... | 14 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $x_{n+2}=\\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\\sum_{n=0}^\\infty\\... | 1685 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCamilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed... | 35 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of three-digit numbers such that:\n- The number formed by the first two digits is pri... | 52 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet the sequence $\\{a_n\\}$ for $n \\ge 0$ be defined as $a_0 = c$, and for $n \\ge 0$, \n$$a_n = \\frac... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nShrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After ... | 173 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 a real number such that $3 \\sin^4 x - 2 \\cos^6 x = -\\frac{17}{25}$. Then, $3 \\cos^4 x - 2 \... | 445 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 answer to this question. What is the value of $2m - 5$?\n\nRemember to put your answer on i... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attache... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 student walks down a hallway with a row of closed lockers numbered from $1$ to $1024$. He opens locker N... | 1024 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nReal numbers $x$, $y$, and $z$ satisfy the following equality: \n\\[4(x+y+z) = x^2 + y^2 + z^2\\] \nLet $M... | 28 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $ax^3 + bx^2 + cx + d = 0$ has three distinct solutions. How many distinct solutions does the... | 2 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $n$ players in a round-robin ping-pong tournament, meaning every two players will play exactly o... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many integers $n$ are there such that $0 \\le n \\le 720$ and $n^2 \\equiv 1 \\pmod{720}$?\n\nRemember... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n12. In a rectangle $ABCD$ with $AB = 8$ and $BC = 20$, let $P$ be a point on $AD$ such that $\\angle BPC =... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ ... | 2037 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, an... | 769 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA number is called *purple* if it can be expressed in the form $\\frac{1}{2^a 5^b}$ for positive integers ... | 109 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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. Determine the smallest possible value of $1 - n + n^2 - n^3 + \\dots + n^{1... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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, x_2, \\dots, x_n$ be a sequence of integers such that $-1 \\leq x_i \\leq 2$ for $i = 1, 2, \\do... | 511 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 row of $100$ cells, each containing a token. For $1$ dollar, it is allowed to interchange two n... | 50 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \\ldots$ Find $n$ such that the first ... | 1027 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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$ be two real numbers such that \\(\\sqrt[3]{a} - \\sqrt[3]{b} = 10\\) and \\(ab = \\left( \\frac... | 280 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 initial number of inhabitants in a city, which is greater than 150, is a perfect square. With an incre... | 249001 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $2018$ ones in a row: $$\\underbrace{1\\,\\,\\,1\\,\\,\\,1\\,\\,\\,1 \\,\\,\\, ... \\,\\,\\,1 \\,\\,... | 1991 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 multiples of $12$ divide $12!$ and have exactly $12$ divisors?\n\nRemember to put your answer on ... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFreddy, the king of flavortext, has an infinite chest of coins. For each number \\( p \\) in the interval ... | 23 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMr. Schwartz has been hired to paint a row of 7 houses. Each house must be painted red, blue, or green. Ho... | 1344 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of addition problems in which a two-digit number is added to a second two-digit number to ... | 3240 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the integer $n$ such that:\n\\[\nn + \\lfloor \\sqrt{n} \\rfloor + \\lfloor \\sqrt{\\sqrt{n}} \\rfloo... | 1967 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $\\triangle ABC$ has radius $5$... | 1814 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 fair six-sided dice are labeled with the numbers $\\{1, 2, 3, 4, 5, 6\\},$ $\\{1, 2, 3, 4, 5, 6\\},$... | 1136 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expres... | 14 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered a... | 1186 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are several teacups in the kitchen, some with handles and others without handles. The number of ways... | 29 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of pairs of integer solutions \\((x,y)\\) that satisfy the equation:\n\\[\n(x-y+2)(x-y-2) ... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the smallest integer $n \\geq 4$ such that $\\binom{n}{4}$ ends in 4 or more zeroes (i.e., the rig... | 8128 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $k$ such that there exist positive integers $a$ and $b$ satisfying the equation... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 prime. It is given that there exists a unique nonconstant function $\\chi:\\{1,2,\\ldots, p-1... | 24 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat’s the smallest integer $n>1$ such that $p \\mid \\left(n^{p-1}-1\\right)$ for all integers $2 \\leq p... | 361 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 201... | -2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe select a real number $\\alpha$ uniformly and at random from the interval $(0,500)$. Define \n\\[ S = \\... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $x, y$ with different parities such that $\\frac{1}{x} + \\frac{1}{y}... | 90 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles $C_1$, $C_2$, $C_3$ have radius $1$ and centers $O$, $P$, $Q$ respectively. $C_1$ and $C_2$ inters... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor what positive integer $k$ is $\\binom{100}{k} \\binom{200}{k}$ maximal?\n\nRemember to put your answer... | 67 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 sequence of eleven squares with side lengths $3, 6, 9, 12, \\ldots, 33$. Eleven copies of a sin... | 414 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree vertices of a parallelogram are \\((2, -4), (-2, 8),\\) and \\((12, 7)\\). Determine the sum of the ... | 12 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\odot$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ ... | 896 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \\[(2^{a+b}+8)... | 62208 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $\\lfloor \\sqrt{19992000} \\rfloor$ where $\\lfloor x \\rfloor$ denotes the greatest integer less th... | 4471 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $A$ lies on the circumference of a circle $\\Omega$ with radius $78$. Point $B$ is placed such that ... | 60 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 all positive integers $N$ for which there exists a unique ordered triple of non-negativ... | 280 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \... | 10053 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCalculate the value of $\\sqrt{\\frac{11^4 + 100^4 + 111^4}{2}}$ and provide the answer as an integer.\n\n... | 11221 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 integer $n$ such that $n$ is divisible by all positive integers less than $\\sqrt[3]{n}$.... | 420 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $AB = AC$. Point $D$ is on side $\\overline{BC}$ so that $AD = CD$ and $\\angle BAD = 3... | 84 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $m$ and $n$ are relatively prime positive integers with $A = \\frac{m}{n}$, where\n\\[ A = \\... | 2015 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $U, S, M, C, A$ are distinct (not necessarily positive) integers such that $U \\cdot S \\cdot M \\cdot ... | 505 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the maximal number of crosses that can fit in a $10 \\times 11$ board without overlapping?\n\nReme... | 14 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ be a real number greater than $1$ such that $\\frac{20a}{a^2+1} = \\sqrt{2}$. Find $\\frac{14a}{a^... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are given that \\(17! = 355687ab8096000\\) for some digits \\(a\\) and \\(b\\). Find the two-digit num... | 42 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least positive integer such that the product of its digits is $8! = 8 \\times 7 \\times 6 \\times... | 257889 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\\overline{BC}$, and denote the circl... | 80 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(x) = x^4 - 4x^3 - 3x^2 - 4x + 1$. Compute the sum of the real roots of $f(x)$.\n\nRemember to put y... | 5 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDarwin takes an $11 \\times 11$ grid of lattice points and connects every pair of points that are 1 unit a... | 220 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integ... | 45 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are given a rectangular playing field of size $13 \\times 2$ and any number of dominoes of sizes $2 \\... | 257 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor real numbers $x$ and $y$, define \\[ \\nabla(x,y)=x-\\frac{1}{y}. \\] If \\[ \\underbrace{\\nabla(2, \... | 203817 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y... | 91 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these ... | 24 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice and Bob play a game together as a team on a $100 \\times 100$ board with all unit squares initially ... | 100 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$,... | 174 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$... | 233 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 two smallest possible values of $x^\\circ$ (in degrees) that satisfy the following equ... | 4064 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $k$ be the smallest positive integer such that the binomial coefficient $\\binom{10^9}{k}$ is less tha... | 38 |
[
{
"role": "user",
"content": "Solve the following math problem step by 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 $f(n) = \\frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all ... | 748 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA and B are friends at a summer school. When B asks A for his address, he answers: \"My house is on XYZ st... | 425 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFreddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n \\ge 1$, Freddy c... | 6423 |
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