messages list | label string | metadata dict | data_source string |
|---|---|---|---|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvery morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a consta... | 204 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 0,
"raw_problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her $4$ hours, including $t$ minutes spent ... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere exist real numbers $x$ and $y$, both greater than 1, such that $$\\log_x(y^x) = \\log_y(x^{4y}) = 10.$$ Find $xy... | 25 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 1,
"raw_problem": "There exist real numbers $x$ and $y$, both greater than 1, such that $$\\log_x(y^x) = \\log_y(x^{4y}) = 10.$$ Find $xy$.\n",
"source_id": null,
"split": null,
"url": null
}
} | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAlice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going... | 809 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 2,
"raw_problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes $1$ token or $4$ tokens from the stack. The player who... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chose... | 116 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 3,
"raw_problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen num... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRectangle $ABCD$ has dimensions $AB = 107$ and $BC = 16$, and rectangle $EFGH$ has dimensions $EF = 184$ and $FG = 17$... | 104 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 4,
"raw_problem": "Rectangle $ABCD$ has dimensions $AB = 107$ and $BC = 16$, and rectangle $EFGH$ has dimensions $EF = 184$ and $FG = 17$. Points $D$, $E$, $C$, and $F$ lie on line $DF$ in that order, and $A$ and $H$ lie on opposite... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\... | 294 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 5,
"raw_problem": "Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\\times 8$ grid. Find the number of such paths that change direction exactly four times, like in... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the largest possible real part of $$(75+117i)z+\\frac{96+144i}{z}$$where $z$ is a complex number with $|z|=4$.\n\... | 540 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 6,
"raw_problem": "Find the largest possible real part of $$(75+117i)z+\\frac{96+144i}{z}$$where $z$ is a complex number with $|z|=4$.\n",
"source_id": null,
"split": null,
"url": null
}
} | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEight circles of radius $34$ can be placed tangent to $\\overline{BC}$ of $\\triangle ABC$ so that the circles are seq... | 197 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 7,
"raw_problem": "Eight circles of radius $34$ can be placed tangent to $\\overline{BC}$ of $\\triangle ABC$ so that the circles are sequentially tangent to each other, with the first circle being tangent to $\\overline{AB}$ and th... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a rhombus whose vertices all lie on the hyperbola $\\tfrac{x^2}{20}-\\tfrac{y^2}{24}=1$ and are in that ... | 480 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 8,
"raw_problem": "Let $ABCD$ be a rhombus whose vertices all lie on the hyperbola $\\tfrac{x^2}{20}-\\tfrac{y^2}{24}=1$ and are in that order. If its diagonals intersect at the origin, find the largest number less than $BD^2$ for a... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point... | 113 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 9,
"raw_problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach vertex of a regular octagon is independently colored either red or blue with equal probability. The probability t... | 371 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 10,
"raw_problem": "Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$. Find the number of intersections of the graphs of ... | 385 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 11,
"raw_problem": "Define $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$. Find the number of intersections of the graphs of $y=4g(f(\\sin (2 \\pi x))) $ and $x=4g(f(\\cos (3 \\pi y)))$.\n",
"source_id": null,
"... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p... | 110 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 12,
"raw_problem": "Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.\n",
... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a tetrahedron such that $AB = CD = \\sqrt{41}$, $AC = BD = \\sqrt{80}$, and $BC = AD = \\sqrt{89}$. Ther... | 104 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 13,
"raw_problem": "Let $ABCD$ be a tetrahedron such that $AB = CD = \\sqrt{41}$, $AC = BD = \\sqrt{80}$, and $BC = AD = \\sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the fa... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ be the radius of th... | 721 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 14,
"raw_problem": "Let $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of $\\mathc... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAmong the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and... | 73 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 15,
"raw_problem": "Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy he... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA list of positive integers has the following properties:\n\n$\\bullet$ The sum of the items in the list is $30$.\n\n$... | 236 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 16,
"raw_problem": "A list of positive integers has the following properties:\n\n$\\bullet$ The sum of the items in the list is $30$.\n\n$\\bullet$ The unique mode of the list is $9$.\n\n$\\bullet$ The median of the list is a positi... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by readi... | 45 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 17,
"raw_problem": "Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n$$\\log_2\\left({x \\over ... | 33 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 18,
"raw_problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n$$\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}$$$$\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}$$$$\\log_2\\l... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCDEF$ be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sid... | 80 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 19,
"raw_problem": "Let $ABCDEF$ be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments $\\overline{AB}$, $\\overline{CD}$, and $\\overline{EF}$ has sid... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAlice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with th... | 55 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 20,
"raw_problem": "Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$. Bob's list has $2024$ sets. Find th... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the re... | 699 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 21,
"raw_problem": "Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectivel... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTorus $T$ is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that i... | 127 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 22,
"raw_problem": "Torus $T$ is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let $S$ be a sphere with a rad... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place ... | 902 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 23,
"raw_problem": "There are $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in a $5 \\times 5$ grid such that\n\n* each cell contains at most one chip... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\triangle$$ABC$ have incenter $I$ and circumcenter $O$ with $\\overline{IA} \\perp \\overline{OI}$, circumradius... | 468 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 24,
"raw_problem": "Let $\\triangle$$ABC$ have incenter $I$ and circumcenter $O$ with $\\overline{IA} \\perp \\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB \\cdot AC$.\n",
"source_id": null,
"split": null,
... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and \n\n$$a^2 b + a^2 c + ... | 601 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 25,
"raw_problem": "Find the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and \n\n$$a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6,000,000.$$\n",
"source_id": null,
"split": null,
... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $O(0,0),A(\\tfrac{1}{2},0),$ and $B(0,\\tfrac{\\sqrt{3}}{2})$ be points in the coordinate plane. Let $\\mathcal{F}... | 23 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 26,
"raw_problem": "Let $O(0,0),A(\\tfrac{1}{2},0),$ and $B(0,\\tfrac{\\sqrt{3}}{2})$ be points in the coordinate plane. Let $\\mathcal{F}$ be the family of segments $\\overline{PQ}$ of unit length lying in the first quadrant with $... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when\n$$\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})$... | 321 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 27,
"raw_problem": "Let $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when\n$$\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})$$\nis divided by 1000.\n",
"source_id": null,
"split": null,
"url": null
}
} | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $b \\geq 2$ be an integer. Call a positive integer $n$ $b\\textit{-eautiful}$ if it has exactly two digits when ex... | 211 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 28,
"raw_problem": "Let $b \\geq 2$ be an integer. Call a positive integer $n$ $b\\textit{-eautiful}$ if it has exactly two digits when expressed in base $b$, and these two digits sum to $\\sqrt{n}$. For example, $81$ is $13$-eautif... | aime-2024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the re... | 315 | {
"ability": "MATH",
"data_source": "aime-2024",
"extra_info": {
"index": 29,
"raw_problem": "Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of... | aime-2024 |
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