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\nThe product $(8)(888\\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What i... | 991 |
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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 $Answer is the answer to the problem.\n\nA \\(4\\times 4\\times h\\) rectangular box contains a sphere of radius \\(2\\) and eight smaller spheres of radius \\(1\\). T... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square... | 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 $Answer is the answer to the problem.\n\nA sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the spher... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are color... | 59 |
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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 $\\triangle BAC$, $\\angle BAC=40^\\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\\overline{AB}$ and $\\overline{... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn rectangle $ABCD$, $\\overline{AB}=20$ and $\\overline{BC}=10$. Let $E$ be a point on $\\overline{CD}$ such that $\\angle CB... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\\leq... | 279 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the spher... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number \\(5^{867}\\) is between \\(2^{2013}\\) and \\(2^{2014}\\). Find how many pairs of integers \\((m,n)\\) there are s... | 279 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a small pond there are eleven lily pads in a row labeled $0$ through $10.$ A frog is sitting on pad $1$. When the frog is o... | 209 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all the positive solutions of \n\\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{2014\\pi^2}{x} \\right) }... | 1080 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMarie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1:00$ PM and finis... | 33 |
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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 Tigers beat the Sharks $2$ out of the $3$ times they played. They then played $N$ more times, and the Sharks ended up winn... | 37 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBack in $1930$, Tillie had to memorize her multiplication facts from $0 \\times 0$ to $12 \\times 12$. The multiplication tabl... | 21 |
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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\nConsider the operation \"minus the reciprocal of,\" defined by $a\\diamond b=a-\\frac{1}{b}$. Find the value of $((1\\diamond2... | 37 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; the... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$, $b$, and $c$ be three distinct one-digit numbers. Find the maximum value of the sum of the roots of the equation $(x-... | 165 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many pairs of parallel edges, such as $\\overline{AB}$ and $\\overline{GH}$ or $\\overline{EH}$ and $\\overline{FG}$, does... | 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 $Answer is the answer to the problem.\n\nThe line \\(12x+5y=60\\) forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this tr... | 294 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many subsets of two elements can be removed from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$ so that the mean (avera... | 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\nLet $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $... | 165 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two si... | 36 |
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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 $Answer is the answer to the problem.\n\nIn a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHexadecimal (base-$16$) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to rep... | 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 $Answer is the answer to the problem.\n\nThe isosceles right triangle \\(ABC\\) has a right angle at \\(C\\) and an area of \\(12.5\\). The rays trisecting \\(\\angle ... | 80 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle ABC$, $\\angle C = 90^\\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle.... | 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\nFor every positive integer $n$, let ${mod}_5 (n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f: ... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. Find the value of $FG + JH + CD$. The original answer is ... | 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\nAaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to th... | -9 |
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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\nFour circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four ci... | 192 |
[
{
"content": "Solve the following math problem step by 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 rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The v... | 10 |
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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 $Answer is the answer to the problem.\n\nA collection of circles in the upper half-plane, all tangent to the \\(x\\)-axis, is constructed in layers as follows. Layer \... | 157 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $... | 2024 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangular box has integer side lengths in the ratio $1: 3: 4$. What could be the volume of the box?\n\nRemember to put you... | 96 |
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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\nWhich of the following numbers is not a perfect square: $1^{2016}$, $2^{2017}$, $3^{2018}$, $4^{2019}$, or $5^{2020}$? The ori... | 2019 |
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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 tens digit of $2015^{2016}-2017?$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": ... | 0 |
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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\n已知 $M=\\left\\{(x, y) \\mid x^{2}+2 y^{2}=3\\right\\}, N=\\{(x, y) \\mid y=m x+b\\}$ 。若对于所有的 $m \\in \\mathbf{R}$, 均有 $M \\cap... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. Th... | 4 |
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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\nCarl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and space... | 336 |
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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 $Answer is the answer to the problem.\n\nHow many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely withi... | 50 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be ... | 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 $Answer is the answer to the problem.\n\nTwo different numbers are randomly selected from the set \\(\\{ -2, -1, 0, 3, 4, 5\\}\\) and multiplied together. What is the ... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers \\(P, Q\\) and \\(R\\), having radii \\(1, 2\\) and \\(3\\), respectively, lie on the same side of line \... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? Provi... | 7 |
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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 $Answer is the answer to the problem.\n\nWhat is the area of the region enclosed by the graph of the equation \\(x^2+y^2=|x|+|y|?\\) Express your answer in the form \\... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRectangle \\(ABCD\\) has \\(AB=5\\) and \\(BC=4\\). Point \\(E\\) lies on \\(\\overline{AB}\\) so that \\(EB=1\\), point \\(G\... | 101 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the pos... | 151 |
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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 binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit... | 109 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadrilateral is inscribed in a circle of radius $200\\sqrt{2}$. Three of the sides of this quadrilateral have length $200$.... | 500 |
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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 $Answer is the answer to the problem.\n\nLet \\(ABCD\\) be a unit square. Let \\(Q_1\\) be the midpoint of \\(\\overline{CD}\\). For \\(i=1,2,\\dots,\\) let \\(P_i\\) ... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree numbers in the interval \\([0,1]\\) are chosen independently and at random. Find the probability that the chosen numbers... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many four-digit integers $abcd$, with $a \\neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an ... | 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\nThere are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\\gcd(a, b, c, d) = 77$ and $\\operatorname{lcm}(a, b... | 27720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the resu... | 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\nSamia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. Th... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe region consisting of all points in three-dimensional space within $3$ units of line segment $\\overline{AB}$ has volume $2... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nClaire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers ... | 143 |
[
{
"content": "Solve the following math problem step by 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 the roots of $z^{12}=64$ that have a positive real part? Express your answer in the form $a\\sqrt{b}+c\\sqr... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn integer $N$ is selected at random in the range $1\\leq N \\leq 2020$. Find the probability that the remainder when $N^{16}$... | 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 $Answer is the answer to the problem.\n\nChloe chooses a real number uniformly at random from the interval \\([0, 2017]\\). Independently, Laurent chooses a real numbe... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be an equilateral triangle. Extend side $\\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3AB$. Similarly, ex... | 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 $Answer is the answer to the problem.\n\nIn the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the mi... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure below, semicircles with centers at \\(A\\) and \\(B\\) and with radii \\(2\\) and \\(1\\), respectively, are dra... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the figure below, choose point \\(D\\) on \\(\\overline{BC}\\) so that \\(\\triangle ACD\\) and \\(\\triangle ABD\\) have e... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integ... | 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\nThe number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. Wh... | 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\nHow many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive,... | 597 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nReal numbers \\(x\\) and \\(y\\) are chosen independently and uniformly at random from the interval \\((0,1)\\). Find the prob... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn integer between $1000$ and $9999$, inclusive, is chosen at random. Find the probability that it is an odd integer whose dig... | 281 |
[
{
"content": "Solve the following math problem step by 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 is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle st... | 39 |
[
{
"content": "Solve the following math problem step by 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 with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are... | 2148 |
[
{
"content": "Solve the following math problem step by 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 $N=123456789101112\\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, o... | 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 $Answer is the answer to the problem.\n\nThe vertices \\(V\\) of a centrally symmetric hexagon in the complex plane are given by \n\\[ V=\\left\\{ \\sqrt{2}i,-\\sqrt{... | 233 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the value of\n\n\\(\\left(\\left((2+1)^{-1}+1\\right)^{-1}+1\\right)^{-1}+1?\\)\n\nProvide your answer in the form of ... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA unit of blood expires after $10!=10\\cdot 9 \\cdot 8 \\cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$... | 22 |
[
{
"content": "Solve the following math problem step by 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 box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement unt... | 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\nIn the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the di... | 19 |
[
{
"content": "Solve the following math problem step by 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 value of \n$$\\log_3 7 \\cdot \\log_5 9 \\cdot \\log_7 11 \\cdot \\log_9 13 \\cdots \\log_{21} 25 \\cdot \\log_{23... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\\overline{FG}$. Find... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \n$$\\begin{align*} x+3y&=3 \\\\ \\... | 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\nA list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct va... | 225 |
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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 $Answer is the answer to the problem.\n\nA paper triangle with sides of lengths \\(3, 4,\\) and \\(5\\) inches, as shown, is folded so that point \\(A\\) falls on poin... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 $a=\\sin 14^{\\circ}+\\cos 14^{\\circ}, b=\\sin 16^{\\circ}+\\cos 16^{\\circ}, c=\\frac{1}{2}\\left(a^{2}+b^{2}\\right)$，请找出... | 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 $Answer is the answer to the problem.\n\nA paper triangle with sides of lengths \\(3, 4,\\) and \\(5\\) inches, as shown, is folded so that point \\(A\\) falls on poin... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\\tria... | 200 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJoey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ ye... | 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\nLet $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?\n\nRemem... | 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\nTwo circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points... | 69 |
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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 $Answer is the answer to the problem.\n\nA closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping pa... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe solutions to the equation \\((z+6)^8=81\\) are connected in the complex plane to form a convex regular polygon, three of w... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nProfessor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. Find th... | 5760 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\... | 340 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA scanning code consists of a $7 \\times 7$ grid of squares, with some of its squares colored black and the rest colored white... | 1022 |
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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\nTriangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn \\(\\triangle{ABC}\\) with side lengths \\(AB = 13\\), \\(AC = 12\\), and \\(BC = 5\\), let \\(O\\) and \\(I\\) denote the ... | 9 |
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