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 $S = \\{1, 2, \\dots, 9\\}$. Compute the number of functions $f : S \\rightarrow S$ such that, for all... | 288 |
[
{
"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\nAn equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \\in \\overline{AB}$, $Q \\in ... | 5073 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the sum of all possible positive integers $n$, such that the product of its digits equals $n^2 -... | 17 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA prime number $p$ is called a \\textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m... | 7 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA group of 6 students decided to form \"study groups\" and \"service activity groups\" based on the follow... | 8 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the... | 2000 |
[
{
"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 positive odd integer that has the same number of positive divisors as $360$?\n\nRemem... | 3465 |
[
{
"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 $f$ be a function such that $f(x) + f(x+1) = 2^x$ and $f(0) = 2010$. Find the last two digits of $f(20... | 51 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}}... | 1973 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such ... | 1958 |
[
{
"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 triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are... | 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\nWhat is the closest number to $169$ that is divisible by $9$?\n\nRemember to put your answer on its own li... | 171 |
[
{
"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 $T = \\{ 1, 2, 3, \\dots, 14, 15 \\}$. A subset $S$ of $T$ is defined as \\textit{handy} if the sum of... | 6560 |
[
{
"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 rectangle with $AB = 6$ and $BC = 6 \\sqrt{3}$. We construct four semicircles $\\omega_1$,... | 243 |
[
{
"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 $\\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Define ... | 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\nThere is a complex number $K$ such that the quadratic polynomial $7x^2 + Kx + 12 - 5i$ has exactly one roo... | 364 |
[
{
"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 fair $6$-sided die is repeatedly rolled until a $1$, $4$, $5$, or $6$ is rolled. What is the expected va... | 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\nLine segment $\\overline{AB}$ has a perpendicular bisector $\\overline{CD}$, where $C$ is the midpoint of ... | 1080 |
[
{
"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 $Answer is the answer to the problem.\n\nLet \\[ f(x) = \\cos(x^3 - 4x^2 + 5x - 2). \\]\nIf we let \\( f^{(k)} \\) denote the \\( k \\)th derivativ... | 907200 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine\n\\[ S = \\tan^{-1}(2020) + \\sum_{j = 0}^{2020} \\tan^{-1}(j^2 - j + 1). \\]\nThen $S$ can be writ... | 4045 |
[
{
"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 $N = \\prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divide... | 624 |
[
{
"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 $Answer is the answer to the problem.\n\nFelix chooses a positive integer as the starting number and writes it on the board. He then repeats the fo... | 21 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \\dots, X_n$ in ... | 4 |
[
{
"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 containing ... | 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\nDetermine the smallest positive integer $M$ with the following property: For every choice of integers $a, ... | 6 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S = \\left\\{ 1,2, \\dots, 2014 \\right\\}$. Suppose that \\[ \\sum_{T \\subseteq S} i^{\\left\\lvert... | 128 |
[
{
"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\nIntegers $a$, $b$, $c$ are selected independently and at random from the set $\\{ 1, 2, \\cdots, 10 \\}$, ... | 136 |
[
{
"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 $a_1, a_2, a_3, \\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n$. Supp... | 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\nHow many ways are there to rearrange the letters of $CCAMB$ such that at least one $C$ comes before the $A... | 40 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $x^5 - 2x^4 - 1 = 0$ has five complex roots $r_1, r_2, r_3, r_4, r_5$. Find the value of \n\\... | 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 $ABCDEF$ be a regular hexagon. Let $P$ be the intersection point of $\\overline{AC}$ and $\\overline{B... | 90 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $$\\sum^{k=672}_{k=0} \\binom{2018}{3k+2} \\pmod{3}$$\n\nRemember to put your answer on its own line ... | 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 $Answer is the answer to the problem.\n\nLet \\[ \\begin{array}{ccccccccccc} A &=& 5 \\cdot 6 &-& 6 \\cdot 7 &+& 7 \\cdot 8 &-& \\cdots &+& 2003 \\... | 20 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo congruent equilateral triangles $\\triangle ABC$ and $\\triangle DEF$ lie on the same side of line $BC... | 33 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relati... | 1360 |
[
{
"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 positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder... | 503 |
[
{
"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\nAll the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in the computer'... | 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\nHarry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve Daily Challenge prob... | 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\nJackson begins at $1$ on the number line. At each step, he remains in place with a probability of $85\\%$ ... | 13446 |
[
{
"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 $Answer is the answer to the problem.\n\nHow many ways are there to color the vertices of a cube using the colors red, blue, or green such that no ... | 114 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how ma... | 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\nLet circle $O$ have radius $5$ with diameter $\\overline{AE}$. Point $F$ is outside circle $O$ such that l... | 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\nFor a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect square... | 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\nCompute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:\n\n- $M... | 4250 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct 3-letter ... | 33 |
[
{
"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 least positive integer $n$ such that $9n$ is a perfect square and $12n$ is a perfect cube?\n\n... | 144 |
[
{
"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\nTwenty-five of the numbers $1, 2, \\ldots, 50$ are chosen. Twenty-five of the numbers $51, 52, \\ldots, 10... | 2525 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe 42 points $P_1, P_2, \\ldots, P_{42}$ lie on a straight line in that order, such that the distance bet... | 861 |
[
{
"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 sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty$ with $n=1,2,...$ has the following property:\n\\[\n... | 40086 |
[
{
"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 box containing $100$ balls, each of which is either orange or black. The box is equally likely ... | 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\nThe cubic polynomials $p(x)$ and $q(x)$ satisfy the following conditions: \n1. $p(1) = q(2)$ \n2. $p(3) ... | 52 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probabil... | 9 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers less than 2015 have exactly 9 positive integer divisors?\n\nRemember to put you... | 13 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in t... | 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\nLet $a$ be a positive real number such that $\\frac{a^2}{a^4-a^2+1}=\\frac{4}{37}$. Then $\\frac{a^3}{a^6-... | 259 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the remainder when \\( \\prod_{n=3}^{33} \\left( 2n^4 - 25n^3 + 33n^2 \\right) \\) is divided by \\( ... | 0 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAndrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was... | 79 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $a$ with $a \\le 154$ are there such that the coefficient of $x^a$ in the expan... | 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\nOn each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \\leq n \\... | 67 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficient... | 4 |
[
{
"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 there exist constants $A$, $B$, $C$, and $D$ such that \n\\[n^4 = A\\binom{n}{4} + B\\binom{n}{3} ... | 75 |
[
{
"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\nAn event occurs periodically over $x$ consecutive years, followed by a break of $y$ consecutive years. The... | 2018 |
[
{
"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 $Answer is the answer to the problem.\n\nSuzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a ro... | 19 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBenji has a $2 \\times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one o... | 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\nLet $a$, $b$, and $c$ be the 3 roots of $x^3 - x + 1 = 0$. Find $\\frac{1}{a+1} + \\frac{1}{b+1} + \\frac{... | -2 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respe... | 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\nLet $\\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possibl... | 640 |
[
{
"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\nBy dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with thi... | 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\nHow many ways are there to fill in a $3 \\times 3$ grid of cells with $0$'s and $2$'s, one number in each ... | 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\nThere is a right triangle $\\triangle ABC$ where $\\angle A$ is the right angle. On side $AB$, there are t... | 277 |
[
{
"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 $\\angle A = 60^\\circ$. Line $\\ell$ intersects segments $AB$ and $AC$ and s... | 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\nGiven positive real numbers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2$ and $ab = c$. Determine the valu... | 4 |
[
{
"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 group of women working together at the same rate can build a wall in $45$ hours. When the work started, ... | 75 |
[
{
"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 trapezoid $ABCD$, where $AD \\parallel BC$ and $\\angle ABC + \\angle CDA = 270^{\\circ}$, compute $AB^... | 260 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAmong $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segme... | 1037 |
[
{
"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 $Answer is the answer to the problem.\n\nPUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose ... | 189 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the following expression:\n\n\\[\n\\frac{2005\\displaystyle \\int_0^{1002}\\frac{dx}{\\sqrt{1002^... | 2005 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the greatest common factor of $12345678987654321$ and $12345654321$?\n\nRemember to put your answe... | 12321 |
[
{
"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 circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\\circ}$. Tw... | 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\nIn triangle $ABC$, $AB = 8$, $AC = 9$, and $BC = 10$. Let $M$ be the midpoint of $BC$. Circle $\\omega_1$ ... | 16295 |
[
{
"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 target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area ... | 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\nFor a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into... | 13 |
[
{
"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 $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find th... | 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\nIf $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values ... | 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\nCalculate $1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 + 9 + \\cdots - 96 + 97 + 98 + 99 + 100$.\n\nRemember to put your... | 202 |
[
{
"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\nIt takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At t... | 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\nFind the smallest positive integer $n$ such that if $n = p_1^{a_1} p_2^{a_2} \\ldots p_s^{a_s}$, then \\[ ... | 59455 |
[
{
"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 sets of $5$ distinct positive integers $A$ satisfy the property that for any positive in... | 4 |
[
{
"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 $Answer is the answer to the problem.\n\nSix chairs are arranged in a row, and six people randomly choose a seat among these chairs. Each person th... | 1106 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that:\n- $f(0) + ... | 7993 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\phi$ be the Euler totient function. Let $\\phi^k (n) = (\\underbrace{\\phi \\circ ... \\circ \\ph... | 6037 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f(n)$ be a function defined on the set of all positive integers and having its values in the same set... | 1988 |
[
{
"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 $Answer is the answer to the problem.\n\n100 schoolchildren participated in the Mathematical Olympiad where 4 tasks were given. The first problem w... | 30 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe unit price of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the tota... | 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\nA sequence $\\{a_n\\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \\ldots, 2020$ and for $i > 2020$,... | 1347 |
[
{
"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 are the last two digits of $2003^{2005^{2007^{2009}}}$, where $a^{b^c}$ means $a^{(b^c)}$?\n\nRemembe... | 43 |
[
{
"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 teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair ... | 666 |
[
{
"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 number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \\equiv 1 \\pmod{20... | 30576 |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the sum of the two smallest positive integers $b$ with the following property: there are at least ... | 540 |
[
{
"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 $x$, $y$, $z$ are real numbers satisfying:\n\\[\n\\begin{align*}\n(x + 1)(y + 1)(z + 1) &= 3, \\\\\n(x ... | 6748 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.