prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by 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 \\[ \\left\\lfloor \\dfrac {2005^3}{2003 \\cdot 2004} - \\dfrac {2003^3}{2004 \\cdot 2005} \\right\\rfloor,\\]where $\... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest two-digit prime that is the sum of three (not necessarily distinct) positive perfect cubes?\n\nRemember t... | 17 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSolve the congruence $6n \\equiv 7 \\pmod{13}$, as a residue modulo 13. (That is, find the value of $n$ that satisfies the co... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A,$ $R,$ $M,$ and $L$ be positive real numbers such that\n\\begin{align*}\n\\log_{10} (AL) + \\log_{10} (AM) &= 2, \\\\\n... | 1000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each is cut by a p... | 32 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA particular fortune cookie lists four two-digit positive integers as your lucky numbers. The first three are 57, 13, and 72, ... | 38 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the product of all real numbers that are doubled when added to their reciprocals?\n\nRemember to put your answer on it... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose the function $f(x)$ is defined on the domain $\\{x_1,x_2,x_3\\}$, so that the graph of $y=f(x)$ consists of just three... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$?\n\nRemember to put y... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right pyramid has a square base with area 288 square cm. Its peak is 15 cm from each of the other vertices. What is the volu... | 864 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $... | 345 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with th... | 336 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are an increasing arithmetic sequence and an ... | 262 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is draw... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of all positive integers that have twice as many digits when written in base $2$ as they have when written in ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$, $b$, and $c$ r... | 358 |
[
{
"content": "Solve the following math problem step by 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 $\\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What... | 99 |
[
{
"content": "Solve the following math problem step by 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 $ABCD$, points $F$ and $G$ lie on $\\overline{AB}$ so that $AF = FG = GB$ and $E$ is the midpoint of $\\overline{... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has center $O,\\ AB=900,\\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\\angle EOF =45^\\ci... | 307 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider all polynomials of the form\n\\[x^9 + a_8 x^8 + a_7 x^7 + \\dots + a_2 x^2 + a_1 x + a_0,\\]where $a_i \\in \\{0,1\\}... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $PQ$ is a straight line, what is the value of $x$? [asy]\n\nsize(150);\n\ndraw((-1,0)--(1,0));\n\ndraw((-0.809,.588)--(0,0)... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the number of digits in the value of $2^{12} \\times 5^8 $.\n\nRemember to put your answer on its own line after \"A... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the tri... | 108 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAssume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of Lena, Wi... | 17700 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the number \\[\\frac{1}{2} \\left(\\frac{5}{\\sqrt[3]{3} + \\sqrt[3]{2}} + \\frac1{\\sqrt[3]{3} -\\sqrt[3]{2}}\\right)\\]ca... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $f(c)=\\frac{3}{2c-3}$, find $\\frac{kn^2}{lm}$ when $f^{-1}(c)\\times c \\times f(c)$ equals the simplified fraction$\\fra... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe volume of the parallelepiped determined by the three-dimensional vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ i... | 80 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRachel has two identical basil plants and an aloe plant. She also has two identical white lamps and two identical red lamps sh... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider two solid spherical balls, one centered at $\\left( 0, 0, \\frac{21}{2} \\right),$ with radius 6, and the other cente... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nQuadrilateral $QABO$ is constructed as shown. Determine the area of $QABO$. [asy]\nsize(5cm);defaultpen(fontsize(9));\npair o ... | 84 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt the Gooddog Obedience School, dogs can learn to do three tricks: sit, stay, and roll over. Of the dogs at the school: \\beg... | 84 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence $\\{a_n\\}$ is defined by $$\na_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\t... | 561 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one t... | 111 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor an arithmetic sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ let\n\\[S_n = a_1 + a_2 + a_3 + \\dots + a_n,\\]and let\n\\[T_n = S_... | 3028 |
[
{
"content": "Solve the following math problem step by 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 $f(x)$ is a function such that\n\\[f(xy) + x = xf(y) + f(x)\\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ t... | 2005 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRegular hexagon $ABCDEF$ is the base of right pyramid $\\allowbreak PABCDEF$. If $PAD$ is an equilateral triangle with side l... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9... | 87 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,$ $b,$ $c,$ $d$ be nonzero integers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^2 = \\begin{pmatrix}... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $\\left\\lfloor n^2/4 \\right\\rfloor - \\lfloor n/2 \\rfloor^2 = 2$, then find all integer values of $n$.\n\nRemember to p... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe product of the proper positive integer factors of $n$ can be written as $n^{(ax+b)/c}$, where $x$ is the number of positiv... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive $3$-digit numbers are multiples of $20$, but not of $55$?\n\nRemember to put your answer on its own line aft... | 41 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA customer ordered 15 pieces of gourmet chocolate. The order can be packaged in small boxes that contain 1, 2 or 4 pieces of c... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind [the decimal form of] the largest prime divisor of $100111011_6$.\n\nRemember to put your answer on its own line after \"... | 181 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEleven girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Ami, starts with the ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\sum_{k = 2}^\\infty \\frac{k - 3}{k(k^2 - 1)}.\\]\n\nRemember to put your answer on its own line after \"Answer:... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose $\\cos V = \\frac{2}{3}$ in the diagram below. What is $TV$?\n\n\n\n[asy]\npair T,U,V;\nV = (0,0);\nT = (0,16);\nU = ... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\[f(x) =\n\\begin{cases}\n|\\lfloor{x}\\rfloor| &\\text{if }x\\text{ is rational}, \\\\\n\\lceil{x}\\rceil^2 &\\text{if }... | 79 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest positive integer $n$ for which $n^2$ is divisible by 18 and $n^3$ is divisible by 640?\n\nRemember to put... | 120 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The numbe... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoint $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$... | 380 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are 4 representatives from each of 4 companies at a convention. At the start of the convention, every person shakes hand... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence $(a_n)$ is defined by $a_1 = 14$ and\n\\[a_n = 24 - 5a_{n - 1}\\]for all $n \\ge 2.$ Then the formula for the $n... | -3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many increasing sequences of three distinct prime numbers have a sum of 20?\n\nRemember to put your answer on its own line... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the b... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly... | 432 |
[
{
"content": "Solve the following math problem step by 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 ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I pai... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence $a_1$, $a_2$, $\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq1$. If $... | 324 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\\angle CDA=60^\\circ$. If the area of $ABCD$ can be written... | 259 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn this Number Wall, you add the numbers next to each other and write the sum in the block directly above the two numbers. Whi... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ represent the smallest integer that satisfies the following conditions:\n$\\frac n2$ is a perfect square.\n$\\frac n3$... | 242 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $BCFE$ is inscribed in right triangle $AGD$, as shown below. If $AB = 28$ units and $CD = 58$ units, what is the area o... | 1624 |
[
{
"content": "Solve the following math problem step by 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 paths are there from $A$ to $B$ on the lines of the grid shown, if every step must be up or to the right?[asy]size(3c... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that\nNone of the first four let... | 320 |
[
{
"content": "Solve the following math problem step by 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 $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ... | 504 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nI want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a di... | 520 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$ Find\n\\[p(1) + p(2) + p(3) + ... | 315 |
[
{
"content": "Solve the following math problem step by 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 triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row ... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\\]for all real numbers $... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer that is both an integer power of 11 and is not a palindrome.\n\nRemember to put your answer... | 161051 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I pai... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\\overl... | 80 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPhoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of ... | 52 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSubtract the number of positive multiples of $3$ that are less than $20$ from the number of positive multiples of $6$ that are... | -3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the are... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid l... | -8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe product of two 2-digit numbers is $3774$. What is the smaller of the two numbers?\n\nRemember to put your answer on its ow... | 51 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied wi... | 105 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $591{,}3d8$ is divisible by 12, what is the sum of all of the digits that could replace $d$?\n\nRemember to put you... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRegular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are copl... | 132 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMatt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins ca... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNotice that \\[35\\cdot40=1400.\\]Find some integer $n$ with $0\\leq n<1399$ such that $n$ is the multiplicative inverse to 1... | 1058 |
[
{
"content": "Solve the following math problem step by 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 of the natural numbers from $1$ to $800,$ inclusive, contain the digit $6$ at least twice? (The numbers $266$ and $66... | 26 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of pairs $(z_1,z_2)$ of complex numbers such that:\n\n$z_1 z_2$ is pure imaginary\n$\\frac{z_1}{z_2}$ is real\... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ cons... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest three-digit integer that is divisible by each of its distinct, non-zero digits.\n\nRemember to put your answe... | 936 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $x^2+14x=33$ has two solutions. The positive solution has the form $\\sqrt{a}-b$ for positive natural numbers $a$... | 89 |
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{
"content": "Solve the following math problem step by 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 coefficient of $x^3$ when $7x^4-3x^3 -3x^2-8x + 1$ is multiplied by $8x^4+2x^3 - 7x^2 + 3x + 4$ and the like terms... | 37 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many solutions does the equation \\[\\frac{(x-1)(x-2)(x-3)\\dotsm(x-100)}{(x-1^2)(x-2^2)(x-3^2)\\dotsm(x-100^2)} = 0\\]hav... | 90 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many times does the digit 9 appear in the list of all integers from 1 to 500? (The number $ 99 $, for example, is counted ... | 100 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe function $f$ is graphed below. Each small box has width and height 1.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspa... | 6 |
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{
"content": "Solve the following math problem step by 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 $\\alpha,$ $\\beta,$ and $\\gamma$ be three real numbers. Suppose that\n\\begin{align*}\n\\cos \\alpha + \\cos \\beta + \... | 12 |
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"content": "Solve the following math problem step by 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 figure, $\\angle ABC$ and $\\angle ADB$ are each right angles. Additionally, $AC = 17.8$ units and $AD = 5$ units. What... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the positive solution to\n\\[\\frac 1{x^2-10x-29}+\\frac1{x^2-10x-45}-\\frac 2{x^2-10x-69}=0.\\]\n\nRemember to put your ... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\\dots,$ $9... | 214 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor some integers that are not palindromes, like 91, a person can create a palindrome by repeatedly reversing the number and a... | 176 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many triangles are in the figure to the right? [asy]\ndefaultpen(linewidth(0.7));\npair hexcoords (real over, real upover)... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose the roots of the polynomial $x^2 - mx + n$ are positive prime integers (not necessarily distinct). Given that $m < 20,... | 18 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many two-digit numbers have digits whose sum is a perfect square?\n\nRemember to put your answer on its own line after \"A... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA book has 136 pages. Each page has the same number of words, and each page has no more than 100 words on it. The number of wo... | 73 |
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