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\nHow many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe area of trapezoid $ABCD$ is $164\\ \\text{cm}^2$. The altitude is $8\\ \\text{cm}$, $AB$ is $10\\ \\text{cm}$, and $CD$ is... | 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 $Answer is the answer to the problem.\n\nThree points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances between... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many two-digit positive integers have at least one $7$ as a digit? Fill in the blank with the correct number of such integ... | 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 $Answer is the answer to the problem.\n\nWhat is the value of \\(x\\) if \\(|x-1|=|x-2|\\)? Provide your answer in the form of \\(\\frac{k}{m}\\), where \\(k\\) and \\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of ... | 26 |
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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 $Answer is the answer to the problem.\n\nAn integer \\(x\\), with \\(10 \\leq x \\leq 99\\), is to be chosen. If all choices are equally likely, what is the probabilit... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token t... | 37 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMinneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Mi... | 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\nA square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. Find the area of the union of... | 175 |
[
{
"content": "Solve the following math problem step by 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 sum of $49$ consecutive integers is $7^5$. What is their median? Provide your answer in the form of $7^x$, where $x$ is an... | 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\nAll the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, a... | 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 $Answer is the answer to the problem.\n\nHenry's Hamburger Haven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pi... | 768 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the United States, coins have the following thicknesses: penny, $1.55 \\, \\text{mm}$; nickel, $1.95 \\, \\text{mm}$; dime,... | 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 $Answer is the answer to the problem.\n\nThirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a b... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe $5\\times 5$ grid shown contains a collection of squares with sizes from $1\\times 1$ to $5\\times 5$. How many of these s... | 19 |
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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 $Answer is the answer to the problem.\n\nThree circles of radius \\(1\\) are externally tangent to each other and internally tangent to a larger circle. The radius of ... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the sequence $2001$, $2002$, $2003$, $\\cdots$, each term after the third is found by subtracting the previous term from th... | 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 $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\), points \\(D\\) and \\(E\\) lie on \\(BC\\) and \\(AC\\), respectively. If \\(AD\\) and \\(BE\\) inter... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSelect numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be th... | 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\nSquare $ABCD$ has side length $2$. A semicircle with diameter $\\overline{AB}$ is constructed inside the square, and the tange... | 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\nThe square \n$$ \n\\begin{array}{|c|c|c|}\\hline50&b&c\\\\\\hline d&e&f\\\\\\hline g&h&2\\\\\\hline\\end{array}\n$$ \ni... | 35 |
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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 $Answer is the answer to the problem.\n\nIn triangle \\(ABC\\) we have \\(AB=7\\), \\(AC=8\\), \\(BC=9\\). Point \\(D\\) is on the circumscribed circle of the triangle... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo \\(4 \\times 4\\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter i... | 30 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this squar... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSquare $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Squa... | 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\nWhat is the area enclosed by the graph of $|3x|+|4y|=12$? Provide your answer as an integer.\n\nRemember to put your answer on... | 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 $Answer is the answer to the problem.\n\nThree tiles are marked \\(X\\) and two other tiles are marked \\(O\\). The five tiles are randomly arranged in a row. Find the... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOne fair die has faces \\(1, 1, 2, 2, 3, 3\\) and another has faces \\(4, 4, 5, 5, 6, 6\\). The dice are rolled and the number... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% ... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers $3, 5, 6, 7$ and $9,$ althoug... | 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 $Answer is the answer to the problem.\n\nOn a standard die, one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. Wha... | 32 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other ... | 96 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without repla... | 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\nThe sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Determine which d... | 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\nA unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure $1$. The cube is then... | 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\nAll of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits... | 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\nOn a certain math exam, $10\\%$ of the students got $70$ points, $25\\%$ got $80$ points, $20\\%$ got $85$ points, $15\\%$ got... | 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 $Answer is the answer to the problem.\n\nA faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer n... | 1462 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\\{-7,-5,-3,-2,2,4,6,13\\}.$ Find the minimum possible value of $... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of complex numbers $z_{0}, z_{1}, z_{2}, \\ldots$ is defined by the rule \n\n$$z_{n+1} = \\frac {iz_{n}}{\\overlin... | 2007 |
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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 $Answer is the answer to the problem.\n\nTwo distinct numbers \\(a\\) and \\(b\\) are chosen randomly from the set \\(\\{2, 2^2, 2^3, \\cdots, 2^{25}\\}\\). Find the p... | 331 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. Find the number of positive integers $n... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously... | 261 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nElisa swims laps in the pool. When she first started, she completed $10$ laps in $25$ minutes. Now, she can finish $12$ laps i... | 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\nMary is about to pay for five items at the grocery store. The prices of the items are $7.99$, $4.99$, $2.99$, $1.99$, and $0.9... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat non-zero real value for \\(x\\) satisfies \\((7x)^{14}=(14x)^7\\)? Provide your answer in the form of \\(\\frac{a}{b}\\),... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many sets of two or more consecutive positive integers have a sum of $15$? Provide your answer as an integer.\n\nRemember ... | 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\nA player pays $\\textdollar 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the numb... | 60 |
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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 $Answer is the answer to the problem.\n\nCassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian l... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(a\\) and \\(b\\) be the roots of the equation \\(x^2-mx+2=0\\). Suppose that \\(a+\\frac{1}{b}\\) and \\(b+\\frac{1}{a}... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other mon... | 30 |
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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 $Answer is the answer to the problem.\n\nSuppose \\(\\cos x=0\\) and \\(\\cos (x+z)=\\frac{1}{2}\\). What is the smallest possible positive value of \\(z\\) in terms o... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers \\(A\\) and \\(B\\) have radius \\(3\\) and \\(8\\), respectively. A common internal tangent intersects t... | 47 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle of radius \\(2\\) is centered at \\(O\\). Square \\(OABC\\) has side length \\(1\\). Sides \\(AB\\) and \\(CB\\) are ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD... | 40400 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers \\(A\\) and \\(B\\) have radius 3 and 8, respectively. A common internal tangent intersects the circles a... | 47 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles with centers \\( O \\) and \\( P \\) have radii \\( 2 \\) and \\( 4 \\), respectively, and are externally tangent. Poi... | 26 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(S\\) be the set of all points \\((x,y)\\) in the coordinate plane such that \\(0 \\le x \\le \\frac{\\pi}{2}\\) and \\(... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many non-empty subsets $S$ of $\\{1,2,3,\\ldots ,15\\}$ have the following two properties? \n\n$(1)$ No two consecutive in... | 405 |
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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 $Answer is the answer to the problem.\n\nA traffic light runs repeatedly through the following cycle: green for \\(30\\) seconds, then yellow for \\(3\\) seconds, and ... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour circles of radius \\(1\\) are each tangent to two sides of a square and externally tangent to a circle of radius \\(2\\),... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe geometric series \\(a+ar+ar^2+\\ldots\\) has a sum of \\(7\\), and the terms involving odd powers of \\(r\\) have a sum of... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\\sqrt{m}$, ... | 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 $a$, $b$, and $c$ be digits with $a\\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a p... | 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 $Answer is the answer to the problem.\n\nA set of 25 square blocks is arranged into a \\(5 \\times 5\\) square. Find the number of different combinations of 3 blocks t... | 600 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by ... | 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\nA lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the... | 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\nA pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of t... | 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\nWhat is the area of the shaded pinwheel shown in the $5 \\times 5$ grid? Provide your answer as an integer.\n\nRemember to put... | 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\nHow many pairs of positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than $1$ and: \n$\... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1,... | 129 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA bakery owner turns on his doughnut machine at $8:30$ AM. At $11:10$ AM the machine has completed one third of the day's job.... | 34 |
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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 $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reverse... | 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 ten-letter code $BEST ~OF ~LUCK$ represents the ten digits $0-9$, in order. What $4$-digit number is represented by the co... | 8671 |
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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 volume of a cube whose surface area is twice that of a cube with volume $1$? Provide your answer in the form of $a... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOlder television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect r... | 27 |
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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 $Answer is the answer to the problem.\n\nPoints \\(A\\) and \\(B\\) are on a circle of radius \\(5\\) and \\(AB = 6\\). Point \\(C\\) is the midpoint of the minor arc ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA ball is dropped from a height of $3$ meters. On its first bounce, it rises to a height of $2$ meters. It keeps falling and b... | 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\nWhat is the area of the region defined by the inequality $|3x-18|+|2y+7|\\le3$?\n\nRemember to put your answer on its own line... | 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\nIn Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer ... | 40 |
[
{
"content": "Solve the following math problem step by 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 floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectang... | 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\nThe numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, a... | 112 |
[
{
"content": "Solve the following math problem step by 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 points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\\t... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTriangle \\(ABC\\), with sides of length \\(5\\), \\(6\\), and \\(7\\), has one vertex on the positive \\(x\\)-axis, one on th... | 95 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRectangle \\(PQRS\\) lies in a plane with \\(PQ=RS=2\\) and \\(QR=SP=6\\). The rectangle is rotated \\(90^\\circ\\) clockwise ... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the expansion of\n\\[\\left(1 + x + x^2 + \\cdots + x^{27}\\right)\\left(1 + x + x^2 + \\cdots + x^{14}\\right)^2,\\]\nfind... | 224 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA function \\(f\\) is defined by \\(f(z) = (4 + i) z^2 + \\alpha z + \\gamma\\) for all complex numbers \\(z\\), where \\(\\al... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTrapezoid $ABCD$ has bases $\\overline{AB}$ and $\\overline{CD}$ and diagonals intersecting at $K.$ Suppose that $AB = 9$, $DC... | 98 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMichael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the p... | 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 $Answer is the answer to the problem.\n\nTwo circles of radius \\(1\\) are to be constructed as follows. The center of circle \\(A\\) is chosen uniformly and at random... | 14 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJacob uses the following procedure to write down a sequence of numbers. First, he chooses the first term to be \\(6\\). To gen... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers?\n\nRemember to put you... | 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\nLet $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\\angle A$ and $\\angle D$ meet at $P$,... | 30 |
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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 $Answer is the answer to the problem.\n\nA carton contains milk that is \\(2\\%\\) fat, an amount that is \\(40\\%\\) less fat than the amount contained in a carton of... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the... | 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\nA particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display ... | 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\nHow many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, ... | 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\nOn Monday, Millie puts a quart of seeds, $25\\%$ of which are millet, into a bird feeder. On each successive day she adds anot... | 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\nPoints \\(A\\) and \\(C\\) lie on a circle centered at \\(O\\), each of \\(\\overline{BA}\\) and \\(\\overline{BC}\\) are tang... | 3 |
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