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\nA grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest di... | 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 $Answer is the answer to the problem.\n\nRight triangle \\(ACD\\) with a right angle at \\(C\\) is constructed outwards on the hypotenuse \\(\\overline{AC}\\) of isosc... | 16 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLine \\(\\ell_1\\) has equation \\(3x-2y=1\\) and goes through \\(A=(-1,-2)\\). Line \\(\\ell_2\\) has equation \\(y=1\\) and ... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(ABCDA'B'C'D'\\) be a rectangular prism with \\(|AB|=2|BC|\\). \\(E\\) is a point on the edge \\([BB']\\) satisfying \\(... | 20 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of triples \\((a,b,c)\\) of positive integers which satisfy the simultaneous equations:\n\\[ ab + bc = 44, \\]... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many subsets of the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}$ are there that do not contain 4 consecutive integers?\n\nRem... | 773 |
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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\nAfter finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average... | 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\nA student is asked to measure $30.0 \\, \\text{g}$ of methanol $(d=0.7914 \\, \\text{g/mL \\text{ at } 25^{\\circ}\\text{C}})$... | 379 |
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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\nTasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square t... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$. Provide yo... | 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\nIf the following instructions are carried out by a computer, what is the value of $X$ that will be printed because of instruct... | 201 |
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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 $Answer is the answer to the problem.\n\nA pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed with... | -2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nJerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis... | 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 $Answer is the answer to the problem.\n\nFive unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending f... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid f... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\(y\\) varies inversely as the square of \\(x\\). When \\(y=16\\), \\(x=1\\). Find the value of \\(y\\) when \\(x=8\\). The o... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose... | 45 |
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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 sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \\ge 2$. Find t... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $D$ be a point on the side $[BC]$ of $\\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\\widehat{ACB})=... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through poi... | 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 $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many dis... | 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 circle $C_1$ with radius $6$ and the circle $C_2$ with radius $8$ are externally tangent to each other at $A$. The circle ... | 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\nWhat is the area enclosed by the graph of $|3x| + |4y| = 12$?\n\nRemember to put your answer on its own line after \"Answer:\"... | 24 |
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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 class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, ... | 24 |
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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 non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons ... | 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\nA regular polygon with $n$ sides is inscribed in a circle with radius $R$. The area of the polygon is $3R^2$. Determine the va... | 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 $Answer is the answer to the problem.\n\nThe sides of a triangle are \\(25, 39,\\) and \\(40\\). What is the diameter of the circumscribed circle? The original answer ... | 128 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the tangents of all interior angles of a triangle are integers, what is the sum of these integers?\n\nRemember to put your ... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAt a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances wit... | -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 $Answer is the answer to the problem.\n\nLet \\(ABC\\) be a triangle with \\(m(\\widehat{A})=90^\\circ\\), \\(|AB|=4\\), and \\(|AC|=3\\). Let \\(D\\) be the foot of t... | 19 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ denote the midpoint of $\\overline{BC}$ and let $E$ denote the... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence $a_1$, $a_2$, $a_3$, $\\dots$ satisfies $a_1 = 19$, $a_9 = 99$, and, for all $n \\ge 3$, $a_n$ is the arithmetic ... | 179 |
[
{
"content": "Solve the following math problem step by 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 $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\\widehat{EAB}) = 11^{\\circ}$, and $m(\\widehat{EBC}) ... | 68 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a circle with center at \\(O\\) and radius \\(r\\), chord \\(AB\\) is drawn with length equal to \\(r\\) (units). From \\(O... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the sum of digits in the decimal representation of a positive integer $n$ is $111$ and the sum of digits in the decimal rep... | 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\nFor how many primes $p$ less than $15$, there exist integer triples $(m,n,k)$ such that:\n\\[\n\\begin{array}{rcl}\nm+n+k &\\e... | 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\nLet $ABCD$ be a convex quadrilateral such that $m (\\widehat{ADB}) = 15^{\\circ}$ and $m (\\widehat{BCD}) = 90^{\\circ}$. The ... | 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 $Answer is the answer to the problem.\n\nThe acronym AMC is shown in the rectangular grid below with grid lines spaced \\(1\\) unit apart. In units, find the sum of th... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$, where $C$ and $D$ are on different semicircles wi... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLarry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a ... | 400 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the value of \\( \\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}} \\). The original answer is in the form \\(\\fr... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles alread... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point object of mass \\(m\\) is connected to a cylinder of radius \\(R\\) via a massless rope. At time \\(t = 0\\), the obje... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(S_1\\) be the area of the regular pentagon \\(ABCDE\\). And let \\(S_2\\) be the area of the regular pentagon whose sid... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle passes through the vertices of a triangle with side-lengths of \\(7\\frac{1}{2}, 10, 12\\frac{1}{2}\\). The radius of... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(V = gt + V_0\\) and \\(S = \\frac{1}{2}gt^2 + V_0t\\), find the expression for \\(t\\) in terms of \\(S\\), \\(V\\), and... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWe draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of... | 40 |
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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 $Answer is the answer to the problem.\n\nHow many pairs of integers \\((x, y)\\) are there such that \\(2x + 5y = xy - 1\\)?\n\nRemember to put your answer on its own ... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFlora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of pr... | 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\nFor how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of ine... | 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 $78$ is divided into three parts which are proportional to $1, \\frac{1}{3}, \\frac{1}{6}$, find the value of the middle pa... | 55 |
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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 $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the nu... | 5720 |
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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 non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times th... | 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\nFor all positive integers $n$, the function $f$ satisfies the following conditions:\n- $f(1) = 1$\n- $f(2n + 1) = 2f(n)$\n- $f... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \\, \\text{cm}$. The length of each side o... | 663 |
[
{
"content": "Solve the following math problem step by 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 consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circ... | 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 $Answer is the answer to the problem.\n\nIn the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers ... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the perimeter of the right triangle whose exradius of the hypotenuse is $30$?\n\nRemember to put your answer on its ow... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\( y = x + \\frac{1}{x} \\), then transform the equation \\( x^4 + x^3 - 4x^2 + x + 1 = 0 \\) into the form \\( x^2(y^2 + ... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Find the value of $\\sin \\theta$. The ori... | 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\nIn $\\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet \\(E\\), \\(F\\), \\(G\\) be points on sides \\([AB]\\), \\([BC]\\), \\([CD]\\) of the rectangle \\(ABCD\\), respectively,... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; a... | 116 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn right triangle \\(ABC\\), \\(BC = 5\\), \\(AC = 12\\), and \\(AM = x\\); \\(\\overline{MN} \\perp \\overline{AC}\\), \\(\\o... | 163 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf the function \\( f \\) satisfies the equation \\( f(x) + f\\left( \\frac{1}{\\sqrt[3]{1-x^3}}\\right) = x^3 \\) for every r... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union... | 175 |
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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\nFigures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping squares, respectively. If the pattern were c... | 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\nRaashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of... | 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\nBy inserting parentheses, it is possible to give the expression $2 \\times 3 + 4 \\times 5$ several values. How many different... | 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 $Answer is the answer to the problem.\n\nA coin is altered so that the probability that it lands on heads is less than \\(\\frac{1}{2}\\). When the coin is flipped fou... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAdams plans a profit of $10\\%$ on the selling price of an article and his expenses are $15\\%$ of sales. If the rate of marku... | 100 |
[
{
"content": "Solve the following math problem step by 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 $2001, 2002, 2003, \\ldots$, each term after the third is found by subtracting the previous term from the sum ... | 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\nConsider the graphs $y = Ax^2$ and $y^2 + 3 = x^2 + 4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. ... | 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\nAt a party with $n \\geq 4$ people, if every 3 people have exactly 1 common friend, find the number of different values that $... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the integer that does not divide $n^{2225} - n^{2005}$ for every integer value of $n$.\n\nRemember to put your answer on ... | 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\nFind the number of primes $p$, such that $x^{3} - 5x^{2} - 22x + 56 \\equiv 0 \\pmod{p}$ has no three distinct integer roots i... | 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\nLet $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\\widehat{EAB}) = 12^\\circ$, and $m(\\widehat{DAE}) ... | 66 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor how many integers \\( n \\) between 1 and 100 does \\( x^2 + x - n \\) factor into the product of two linear factors with ... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the greatest integer $k$ which makes the statement \"When we take any $6$ subsets with $5$ elements of the set \\( \\{... | 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\nSuppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\\overline{ab}$ is expresse... | 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\nTwo six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ rep... | 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 $Answer is the answer to the problem.\n\nLet \\(a\\), \\(b\\), and \\(c\\) be the roots of the polynomial \\(x^3 + 4x^2 - 7x - 1\\). Find the polynomial that has roots... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral with $\\angle C = \\angle D = 90^\\circ$. The circle $K$ passing through $A$ and $B$ is t... | 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\nKate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?\n\nRemember to put your ans... | 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\nOn circle $O$, points $C$ and $D$ are on the same side of diameter $\\overline{AB}$, $\\angle AOC = 30^\\circ$, and $\\angle D... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a right triangle with $m(\\widehat{A})=90^\\circ$. Let $APQR$ be a square with area $9$ such that $P \\in [AC]$, ... | 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 $Answer is the answer to the problem.\n\nLet \\(D\\) be a point on side \\([BC]\\) of \\(\\triangle ABC\\) such that \\(|AB|=3\\), \\(|CD|=1\\), and \\(|AC|=|BD|=\\sqr... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhen $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as ... | 337 |
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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\nGiven three squares as in the figure, where the largest square has area 1, and the area $A$ is known. Find the area $B$ of the... | 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\nIf $a$ and $b$ are the roots of the equation $x^2 - 2cx - 5d = 0$, and $c$ and $d$ are the roots of the equation $x^2 - 2ax - ... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf \\(x \\neq 0\\) or \\(4\\) and \\(y \\neq 0\\) or \\(6\\), then \\(\\frac{2}{x}+\\frac{3}{y}=\\frac{1}{2}\\) is equivalent ... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \\sqrt{15}$ square inches. Find the... | 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\nFor every $n \\ge 2$, $a_n = \\frac{\\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying... | 106 |
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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 pattern of figures $\\triangle$ $\\bullet$ $\\square$ $\\blacktriangle$ $\\circ$ is repeated in the sequence:\n\\[\n\\tria... | 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 $Answer is the answer to the problem.\n\nWhat is the sum of distinct real numbers \\(x\\) such that \\((2x^2+5x+9)^2=56(x^3+1)\\)? The original answer is in \\(\\frac{... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ordered triples of integers $(a, b, c)$ satisfy the following equations?\n\\[ |a + b| + c = 19 \\]\n\\[ ab + |c| = 97... | 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 $Answer is the answer to the problem.\n\nThere are integers \\( a \\), \\( b \\), and \\( c \\), each greater than 1, such that \\( \\sqrt[a]{N \\sqrt[b]{N \\sqrt[c]{N... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the sum of the real roots of the equation $(x\\sqrt{x})^x = x^{x\\sqrt{x}}$? Provide your answer in the form of a simp... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nMenkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have a... | 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\nEmily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, a... | 9 |
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