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\nLet $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\\sqrt{6}$, $m(\\widehat{ABD})=60^\\circ$, $m(\\widehat{BDC})... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe zeroes of the function $f(x) = x^2 - ax + 2a$ are integers. Find the sum of all possible values of $a$.\n\nRemember to put... | 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\nLet $f(x) = x - \\frac{1}{x}$. How many different solutions are there to the equation $f(f(f(x))) = 1$? Provide your answer as... | 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 point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0)$, $(2020, 0)$, $(2020, 2020)$... | 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\nLines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\[x = .123456789101112\\ldots998999,\\] where the digits are obtained by writing the integers 1 through 999 in order. Wha... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCalculate the value of \\(\\dfrac{3\\times 5}{9\\times 11}\\times \\dfrac{7\\times 9\\times 11}{3\\times 5\\times 7}\\).\n\nRe... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many whole numbers between $100$ and $400$ contain the digit $2$? Fill in the blank with the number of such whole numbers.... | 138 |
[
{
"content": "Solve the following math problem step by 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 grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence ... | 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\nThe large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is s... | 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\nIf $A$ and $B$ are nonzero digits, then find the number of digits (not necessarily different) in the sum of the three whole nu... | 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 $Answer is the answer to the problem.\n\nIf rose bushes are spaced about 1 foot apart, approximately how many bushes are needed to surround a circular patio whose radi... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA fifth number, $n$, is added to the set $\\{ 3,6,9,10 \\}$ to make the mean of the set of five numbers equal to its median. F... | 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\nWhen placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, find the smallest possibl... | 149 |
[
{
"content": "Solve the following math problem step by 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 $47$ be written as the sum of two primes?\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 $Answer is the answer to the problem.\n\nEvery time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the tw... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one ... | 1047 |
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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 decimal number $0.12345$, which digit, when changed to $9$, gives the largest possible number? Indicate the original po... | 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 $Answer is the answer to the problem.\n\nIf an integer of two digits is \\(k\\) times the sum of its digits, find the expression that represents the number formed by i... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSeveral students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person re... | 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 $Answer is the answer to the problem.\n\nHow many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest quotient that can be formed using two numbers chosen from the set $\\{ -24, -3, -2, 1, 2, 8 \\}$?\n\nRemem... | 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\nA cube of edge $3 \\text{ cm}$ is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes ... | 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\nThe digit-sum of $998$ is $9+9+8=26$. Find the number of $3$-digit whole numbers, whose digit-sum is $26$, that are even.\n\nR... | 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\nThe distance between the $5^\\text{th}$ and $26^\\text{th}$ exits on an interstate highway is $118$ miles. If any two consecut... | 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\nThe rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$,... | 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\nCalculate the value of $(1901+1902+1903+\\cdots + 1993) - (101+102+103+\\cdots + 193)$.\n\nRemember to put your answer on its ... | 167400 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhen $10^{93}-93$ is expressed as a single whole number, what is the sum of the digits?\n\nRemember to put your answer on its ... | 826 |
[
{
"content": "Solve the following math problem step by 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 children at a time can play pairball. For $90$ minutes, with only two children playing at a time, five children take turns... | 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 $Answer is the answer to the problem.\n\nLet \\(W, X, Y,\\) and \\(Z\\) be four different digits selected from the set \n\n\\(\\{ 1,2,3,4,5,6,7,8,9\\}.\\) \n\nFind t... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. I... | 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 $Answer is the answer to the problem.\n\nFind the sum of the digits in the answer to \n \n$\\underbrace{9999\\cdots 99}_{94\\mathrm{ nines}} \\times \\underbrace{444... | 846 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhich operation has the same effect on a number as multiplying by \\(\\dfrac{3}{4}\\) and then dividing by \\(\\dfrac{3}{5}\\)... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSix different digits from the set \n$\\{ 1,2,3,4,5,6,7,8,9\\}$ \nare placed in the squares in the figure shown so that the su... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many subsets containing three different numbers can be selected from the set \n$\\{ 89,95,99,132,166,173 \\}$ \nso that ... | 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\nAhn chooses a two-digit integer, subtracts it from $200$, and doubles the result. What is the largest number Ahn can get? Prov... | 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 $Answer is the answer to the problem.\n\nEach side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square ... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are positive integers that have these properties: \n \nthe sum of the squares of their digits is $50$, and \neach dig... | 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\nAll of the even numbers from $2$ to $98$ inclusive, excluding those ending in $0,$ are multiplied together. What is the rightm... | 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 $Answer is the answer to the problem.\n\nCalculate the value of \\(100 \\times 19.98 \\times 1.998 \\times 1000\\) and express your answer in the form of \\(x^2\\). Wh... | 1998 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTerri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the ... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangular board of $8$ columns has squares numbered beginning in the upper left corner and moving left to right so row one... | 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\nEach of the five numbers $1, 4, 7, 10$, and $13$ is placed in one of the five squares so that the sum of the three numbers in ... | 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 $Answer is the answer to the problem.\n\nIn a far-off land, three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. ... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides o... | 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 year $2001$, the United States will host the International Mathematical Olympiad. Let $I, M,$ and $O$ be distinct posit... | 671 |
[
{
"content": "Solve the following math problem step by 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 $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\\triangle PAB$. As $P$ moves along a line that is parallel to... | 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\nIf $|x - 2| = p$, where $x < 2$, then find the value of $x - p$. The original answer is in the form $a - bp$, please provide t... | 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 $Answer is the answer to the problem.\n\nIn rectangle \\(ABCD\\), \\(AD=1\\), \\(P\\) is on \\(\\overline{AB}\\), and \\(\\overline{DB}\\) and \\(\\overline{DP}\\) tri... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFigures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern w... | 20201 |
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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 $Answer is the answer to the problem.\n\nMrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, w... | 80 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of \\( A \\cdot M \\cdot C + A \\c... | 112 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a function for which $f\\left(\\dfrac{x}{3}\\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that ... | 555 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle centered at \\(O\\) has radius \\(1\\) and contains the point \\(A\\). The segment \\(AB\\) is tangent to the circle ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThrough a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triang... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\\overline{BC}$ and let $E$ denote the int... | 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 $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. Find the maximum value of $A \\cdot M \\cdot C + A \\cdot ... | 69 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nKeiko tosses one penny and Ephraim tosses two pennies. Find the probability that Ephraim gets the same number of heads that Ke... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe area of rectangle $ABCD$ is $72$ units squared. If point $A$ and the midpoints of $\\overline{BC}$ and $\\overline{CD}$ ar... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?\n\nRemember to put your ans... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The spee... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA charity sells $140$ benefit tickets for a total of $＄2001$. Some tickets sell for full price (a whole dollar amount), and th... | 782 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an integer.\n\... | 801 |
[
{
"content": "Solve the following math problem step by 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 street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The l... | 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 $Answer is the answer to the problem.\n\nA square piece of paper, \\(4\\) inches on a side, is folded in half vertically. Both layers are then cut in half parallel to ... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter \... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFour positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy: \n \n$$\\begin{array}{rl} ab + a + b & = 5... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoints $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How m... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $24$ four-digit whole numbers that use each of the four digits $2, 4, 5$ and $7$ exactly once. Only one of these fou... | 7425 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe function $f$ is given by the table \n$$\\begin{array}{|c||c|c|c|c|c|}\\hline\\text{x}&1&2&3&4&5\\\\\\hline\\text{f(x)}&4&... | 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 arithmetic mean of the nine numbers in the set $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$ is a $9$-digit number $M$, all... | 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 $Answer is the answer to the problem.\n\nEach of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, a... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the... | 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 $Answer is the answer to the problem.\n\nFind \\(i + 2i^2 + 3i^3 + \\cdots + 2002i^{2002}\\). The original answer is in the form \\(a + bi\\), where \\(a\\) and \\(b\\... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUsing the digits $1, 2, 3, 4, 5, 6, 7,$ and $9,$ form $4$ two-digit prime numbers, using each digit only once. What is the sum... | 190 |
[
{
"content": "Solve the following math problem step by 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 $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random ... | 2002 |
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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 $3\\times3\\times3$ cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible su... | 90 |
[
{
"content": "Solve the following math problem step by 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 integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number?\n\nRemember to put your answer on its own line a... | 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 $Answer is the answer to the problem.\n\nSpot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard r... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $\\{a_n\\}$ is an arithmetic sequence with \n$$a_1+a_2+\\cdots+a_{100}=100 ~\\text{and}~ a_{101}+a_{102}+\\cdots... | 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\nLet $\\triangle XOY$ be a right-angled triangle with $m\\angle XOY = 90^{\\circ}$. Let $M$ and $N$ be the midpoints of legs $O... | 26 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider the sequence of numbers: $4,7,1,8,9,7,6,\\cdots$ For $n>2$, the $n$-th term of the sequence is the units digit of the... | 1999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nUnder the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unans... | 1885 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all integers $n$ greater than $1$, define $a_n = \\frac{1}{\\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10}... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation \\(z(z+i)(z+3i)=2002i\\) has a zero of the form \\(a+bi\\), where \\(a\\) and \\(b\\) are positive real numbers. ... | 118 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\), we have \\(AB = 1\\) and \\(AC = 2\\). Side \\(\\overline{BC}\\) and the median from \\(A\\) to \\(\\... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$. Find the area of $ABCD$.\... | 210 |
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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 $Answer is the answer to the problem.\n\nLoki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gav... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is the value of $A+M+C$?\n\nRemember to put your an... | 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 symbolism $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. For example, $\\lfloor 3 \\rfloor = 3,$ and... | 38 |
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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 $Answer is the answer to the problem.\n\nSimplify \\(\\sqrt[3]{x\\sqrt[3]{x\\sqrt[3]{x\\sqrt{x}}}}\\) and express your answer in the form of \\(\\sqrt[k]{x^p}\\). Plea... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four... | 36 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAl, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of \... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that t... | 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 $Answer is the answer to the problem.\n\nA regular octagon \\(ABCDEFGH\\) has an area of one square unit. What is the area of the rectangle \\(ABEF\\)? Provide your an... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point \\(P\\) is chosen at random in the interior of equilateral triangle \\(ABC\\). What is the probability that \\(\\trian... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(\\log (xy^3) = 1\\) and \\(\\log (x^2y) = 1\\), find the value of \\(\\log (xy)\\). The answer should be in the form \\(... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn rectangle \\(ABCD\\), \\(AB=5\\) and \\(BC=3\\). Points \\(F\\) and \\(G\\) are on \\(\\overline{CD}\\) so that \\(DF=1\\) ... | 27 |
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