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\nIn triangle $ABC$, angle $B$ is obtuse and equal to $a$. The bisectors of angles $A$ and $C$ intersect the... | 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\nA tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. I... | 252 |
[
{
"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\nPositive rational numbers $x < y < z$ sum to $1$ and satisfy the equation:\n\\[ (x^2 + y^2 + z^2 - 1)^3 + ... | 999 |
[
{
"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 disk with radius $10$ and a disk with radius $8$ are drawn such that the distance between their centers ... | 19 |
[
{
"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 smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \\ldots < ... | 2024 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine a search algorithm called $\\texttt{powSearch}$. Assume $A$ is a 1-indexed sorted array of distinct... | 808 |
[
{
"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 $Answer is the answer to the problem.\n\nThe sequence of digits \\(1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \\dots\\) is o... | 1984 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the... | 4807 |
[
{
"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 positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the sma... | 120 |
[
{
"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\nQuadrilateral $ABCD$ satisfies the following conditions: \n- $AB = 8$, \n- $BC = 5$, \n- $CD = 17$, \n- $D... | 60 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such ... | 500 |
[
{
"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 $f_1(x) = x^2 - 1$, and for each positive integer $n \\geq 2$, define $f_n(x) = f_{n-1}(f_1(x))$. How ... | 2005 |
[
{
"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\n30 segments of lengths $1, \\sqrt{3}, \\sqrt{5}, \\sqrt{7}, \\sqrt{9}, \\ldots, \\sqrt{59}$ have been draw... | 30 |
[
{
"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 $Answer is the answer to the problem.\n\nA large number of rocks are placed on a table. On each turn, one may remove some rocks from the table foll... | 18 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt a movie theater, tickets for adults cost $4 more than tickets for children. One afternoon, the theater ... | 1875 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt a party, every guest is a friend of exactly fourteen other guests (not including themselves). Every two... | 64 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a right trapezoid $ABCD$ where $AB \\parallel CD$, the angle at vertex $B$ measures $75^\\circ$. Point ... | 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\nFind the sum of all the digits in the decimal representations of all the positive integers less than $1000... | 13500 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the pla... | 1247 |
[
{
"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 $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality ... | 256 |
[
{
"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 $a = \\sqrt[401]{4} - 1$ and for each $n \\ge 2$, let $b_n = \\binom{n}{1} + \\binom{n}{2} a + \\ldots... | 1024 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$... | 18 |
[
{
"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 $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $... | 1112 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\alpha, \\beta$ be the distinct positive roots of the equation $2x = \\tan x$. Evaluate the integral... | 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 $Answer is the answer to the problem.\n\nWe are given 5040 balls in \\(k\\) different colors, where the number of balls of each color is the same. ... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $2008 \\times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of sq... | 4068434225296 |
[
{
"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\nQuadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its side lengths are integers, $A... | 30 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe volume of a certain rectangular solid is $216\\text{ cm}^3$, its total surface area is $288\\text{ cm}... | 96 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, define a function $f_n(x)$ on the interval $[0, n+1]$ as follows:\n\\[ f_n(x) ... | 450 |
[
{
"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 $ABCD$ be a trapezoid with $AB \\parallel CD$, $AB = 5$, $BC = 9$, $CD = 10$, and $DA = 7$. Lines $BC$... | 90011 |
[
{
"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 value of $(a+5)^2 + (b-2)^2 + (c-9)^2$ for all real numbers $a, b, c$ satisfying $a... | 72 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $u_n$ be the $n^{\\text{th}}$ term of the sequence:\n\\[\n1, \\quad 2, \\quad 5, \\quad 6, \\quad 9, \... | 5898 |
[
{
"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 $Answer is the answer to the problem.\n\nFind the number of ordered pairs \\((a, b)\\) of positive integers that are solutions of the following equ... | 1 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Comput... | 849 |
[
{
"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 $Answer is the answer to the problem.\n\nJim has two fair 6-sided dice: one die has faces labelled from 1 to 6, and the second die has faces labell... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p > \\sqrt{n}$... | 69 |
[
{
"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 real numbers $x$, $y$, $z$, and $t$ satisfy the following equation:\n\\[ 2x^2 + 4xy + 3y^2 - 2xz - 2yz... | 125 |
[
{
"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 functions $f : \\{1, 2, \\ldots, 16\\} \\to \\{1, 2, \\ldots, 16\\}$ have the property that $f(f(... | 16 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDaniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two p... | 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\nIn the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\\frac{p... | 3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.\n\nRemember to put your answer on its ow... | 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 $Answer is the answer to the problem.\n\nWhat is the smallest integer \\( a \\) greater than 14 such that a triangle with side lengths \\( a - 1 \\... | 52 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rhombus with sidelength $1$ has an inscribed circle with radius $\\frac{1}{3}$. If the area of the rhomb... | 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\nWhat is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two num... | 987654301 |
[
{
"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 triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\\triangle ABC$ is centered at ... | 720 |
[
{
"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 $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$... | 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\nConsider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the... | 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\nLet $O$, $A$, $B$, and $C$ be points in space such that $\\angle AOB = 60^{\\circ}$, $\\angle BOC = 90^{\\... | 109 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such th... | 30 |
[
{
"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 the smallest integer such that the sum of digits of $n$ is divisible by $5$, and the sum of dig... | 49 |
[
{
"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 two-digit number $N$ which is divisible by 4, such that all integral powers of $N$ end wi... | 76 |
[
{
"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\nAidan rolls a pair of fair, six-sided dice. Let $n$ be the probability that the product of the two numbers... | 7 |
[
{
"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 a constant such that exactly three real values of $x$ satisfy $$x - |x^2 - 4x + 3| = k.$$ The s... | 11 |
[
{
"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 $x, y \\in \\mathbb{R}$ such that $x > 2, y > 3$. Find the minimum value of \\( \\frac{(x+y)^2}{\\sqrt... | 10 |
[
{
"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 $Answer is the answer to the problem.\n\n$p, q, r$ are real numbers satisfying the following equations: \n\\[ \\frac{(p+q)(q+r)(r+p)}{pqr} = 24 \\... | 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\nLet $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is sh... | 126 |
[
{
"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 two real numbers $a$ and $b$ that satisfy the equations:\n\\[\na^4 + a^2b^2 + b^4 = 900,\n\\]\n\\[\n... | 25 |
[
{
"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 integer solutions exist that satisfy the equation?\n\n$$x + 4y - 343\\sqrt{x} - 686\\sqrt{y} + 4\... | 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\nGiven two non-negative integers $m > n$, let's say that $m$ ends in $n$ if we can obtain $n$ by erasing so... | 103 |
[
{
"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 sets of positive integers \\((a, b, c)\\) satisfy \\(a > b > c > 0\\) and \\(a + b + c = 103\\)?\... | 833 |
[
{
"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 triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\\math... | 96 |
[
{
"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 $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \\perp BC$. Let $\\ell_1... | 113 |
[
{
"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 largest two-digit integer for which the product of its digits is $17$ more than their sum?\n\n... | 74 |
[
{
"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 unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\\overline... | 7 |
[
{
"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\nEight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probabilit... | 843 |
[
{
"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 distinct four-letter arrangements can be formed by rearranging the letters found in the word **FL... | 72 |
[
{
"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 mason has bricks with dimensions $2 \\times 5 \\times 8$ and other bricks with dimensions $2 \\times 3 \... | 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\nLet $\\{a_n\\}$ be a sequence of integers such that $a_1=2016$ and \\[\\frac{a_{n-1}+a_n}{2}=n^2-n+1\\] fo... | 7986 |
[
{
"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_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, \\dots, $a_9 = 9^{a_8}$. Find the last two digit... | 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\nSquare trinomial $f(x)$ is such that the polynomial $f(x)^5 - f(x)$ has exactly three real roots. Find the... | 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\nThe following number is the product of the divisors of $n$:\n$$46,656,000,000$$\nWhat is $n$?\n\nRemember ... | 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\nDefine $\\{p_n\\}_{n=0}^\\infty\\subset\\mathbb{N}$ and $\\{q_n\\}_{n=0}^\\infty\\subset\\mathbb{N}$ to be... | 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\nLet $N$ be the number of functions $f: \\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have ... | 31 |
[
{
"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 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers ... | 39 |
[
{
"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 $Answer is the answer to the problem.\n\nWhat is the largest number of points that can be arranged on a given circle such that in all the triangles... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \\[ N = 1111\\cdots 11. ... | 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 $BCDE$ be a trapezoid with $BE \\parallel CD$, $BE = 20$, $BC = 2\\sqrt{34}$, $CD = 8$, $DE = 2\\sqrt{... | 203 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute $\\sum_{n=1}^{\\infty}\\frac{n + 1}{n^2(n + 2)^2}$.\n\nExpress your answer in simplest form as $\\... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $M$ be the set of all polynomials $P(x)$ with pairwise distinct integer roots, integer coefficients, a... | 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 $Answer is the answer to the problem.\n\nCompute the largest integer \\( N \\leq 2012 \\) with four distinct digits.\n\nRemember to put your answer... | 1987 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA competition room of HOMC has $m \\times n$ students where $m, n$ are integers larger than $2$. Their sea... | 18 |
[
{
"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\nPoints $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on ... | 47 |
[
{
"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 Occidentalia, there are $20$ different companies, each looking to hire $15$ new employees. A group of $... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any real number $x$, let $\\lfloor x \\rfloor$ denote the integer part of $x$; $\\{ x \\}$ be the frac... | 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\nThe area of the largest square that can be inscribed in a regular hexagon with side length $1$ can be expr... | 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\nConsider two geometric sequences $16$, $a_1$, $a_2$, $\\ldots$ and $56$, $b_1$, $b_2$, $\\ldots$ with the ... | 490 |
[
{
"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 $Answer is the answer to the problem.\n\nHow many positive integer solutions does the equation have:\n\\[ \\left\\lfloor \\frac{x}{10} \\right\\rfl... | 110 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDoes there exist a four-digit positive integer with different non-zero digits, which has the following pro... | 1234 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\\ell$ passing through $B... | 157 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $5$ dogs, $4$ cats, and $7$ bowls of milk at an animal gathering. Dogs and cats are distinguisha... | 20 |
[
{
"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 $Answer is the answer to the problem.\n\nLet \\( f(x) \\) be a monic cubic polynomial with \\( f(0) = -64 \\), and all roots of \\( f(x) \\) are no... | -125 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat number is nine more than four times the answer to this question?\n\nRemember to put your answer on it... | -3 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant spe... | 210 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the rem... | 693 |
[
{
"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 ways are there to permute the letters $\\{S, C, R, A, M, B, L, E\\}$ without the permutation cont... | 40200 |
[
{
"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 $Answer is the answer to the problem.\n\n\\( f \\) is a function whose domain is the set of nonnegative integers and whose range is contained in th... | 2015 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all positive integers $n$, denote by $\\sigma(n)$ the sum of the positive divisors of $n$ and $\\nu_p(... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_n = 2^{2^n} + 1$ and let $m$ be the least common multiple of $x_2, x_3, \\ldots, x_{1971}$. Find th... | 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\nJuan makes a list of $2018$ numbers. The first number is $1$. Each subsequent number is obtained by adding... | 17929 |
[
{
"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 three-digit number that satisfies the following properties:\n- The hundreds digit and ones dig... | 171 |
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