messages listlengths 1 1 | label stringlengths 1 8 | metadata dict | data_source stringclasses 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: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAn explorer is challenging to pass through a maze composed of 3033 rows and 3032 columns of squares. The rules of the ... | 3 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 90,
"raw_problem": "An explorer is challenging to pass through a maze composed of 3033 rows and 3032 columns of squares. The rules of the maze are as follows: In all the middle rows (that is, from the 2nd row to the 3032nd row), t... | aime-beyond |
[
{
"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\nFor a positive integer \\(n\\geq2\\)，define \\(n - 1\\) by \\(n-1\\) matrix \\(M_n(\\lambda)=\\begin{pmatrix}6-\\lambd... | 4 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 91,
"raw_problem": "For a positive integer \\(n\\geq2\\)，define \\(n - 1\\) by \\(n-1\\) matrix \\(M_n(\\lambda)=\\begin{pmatrix}6-\\lambda&-\\lambda&-\\lambda&\\cdots&-\\lambda\\\\-\\lambda&24-\\lambda&-\\lambda&\\cdots&-\\lambda... | aime-beyond |
[
{
"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 sequence of real numbers \\(\\{a_n\\}\\) satisfies that：\\(a_{n + 1}=2^n-7a_n，n = 0,1,2,\\cdots\\). Find the minimal... | 9 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 92,
"raw_problem": "A sequence of real numbers \\(\\{a_n\\}\\) satisfies that：\\(a_{n + 1}=2^n-7a_n，n = 0,1,2,\\cdots\\). Find the minimal possible value of \\(\\frac{1}{a_0}\\) such that \\(a_{n + 1}>a_n\\) for any positive integ... | aime-beyond |
[
{
"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 frog starts from 0 and makes 2047 jumps. This process satisfies the following conditions: (1) The frog will jump to ... | 11 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 93,
"raw_problem": "A frog starts from 0 and makes 2047 jumps. This process satisfies the following conditions: (1) The frog will jump to each point in the set \\(\\{1, 2, 3,\\cdots,2047\\}\\) exactly once without repetition or om... | aime-beyond |
[
{
"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\nFor any positive integer \\(n\\geq2\\), we define \\(f(n)\\) as the last non-zero digit of \\(n!\\) in the decimal sys... | 12 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 94,
"raw_problem": "For any positive integer \\(n\\geq2\\), we define \\(f(n)\\) as the last non-zero digit of \\(n!\\) in the decimal system. It is known that \\(k\\) is the fifth smallest positive integer that satisfies the foll... | aime-beyond |
[
{
"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\nHow many integers \\(k\\in[2,18676]\\) are there such that the number of positive integer triples \\((n,a,b)\\) satisf... | 14 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 95,
"raw_problem": "How many integers \\(k\\in[2,18676]\\) are there such that the number of positive integer triples \\((n,a,b)\\) satisfying the following conditions is finite: \\(n\\) is odd, \\(\\gcd(a,b) = 1\\), \\(a + b=k\\)... | aime-beyond |
[
{
"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 60 real numbers \\(a_1, a_2, \\cdots, a_{60}\\) satisfying \\(a_1 + a_2+\\cdots + a_{60}=0\\), and for any \... | 15 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 96,
"raw_problem": "There are 60 real numbers \\(a_1, a_2, \\cdots, a_{60}\\) satisfying \\(a_1 + a_2+\\cdots + a_{60}=0\\), and for any \\(1\\leqslant i\\leqslant 60\\), \\(\\vert a_i - a_{i + 1}\\vert\\leqslant 1\\) (\\(a_{61}=a... | aime-beyond |
[
{
"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\nAt least how many black stones should be placed on a 9×9 square grid board to ensure that there is no possibility for ... | 16 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 97,
"raw_problem": "At least how many black stones should be placed on a 9×9 square grid board to ensure that there is no possibility for the white to make a five-in-a-row? (which means there are no five consecutive empty squares ... | aime-beyond |
[
{
"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\n\\(n\\) students take an exam. There are a total of 6 questions in the exam, and each question has three options. Afte... | 18 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 98,
"raw_problem": "\\(n\\) students take an exam. There are a total of 6 questions in the exam, and each question has three options. After the exam, it is statistically found that for any two students, the number of questions wit... | aime-beyond |
[
{
"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\nPlace \\(m\\) \\(1 \\times 2\\) dominos on a \\(7\\times7\\) chessboard such that no domino can slide horizontally or ... | 22 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 99,
"raw_problem": "Place \\(m\\) \\(1 \\times 2\\) dominos on a \\(7\\times7\\) chessboard such that no domino can slide horizontally or vertically when other dominos remains fixed. Find the minimal possible value of \\(m\\).",
... | aime-beyond |
[
{
"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 minimal positive integer which can not be represented as \\(\\sum_{i = 1}^{7263}(- 1)^{a_{i}}\\time... | 23 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 100,
"raw_problem": "Let \\(N\\) be the minimal positive integer which can not be represented as \\(\\sum_{i = 1}^{7263}(- 1)^{a_{i}}\\times2^{b_{i}}\\) where \\(a_{i},b_{i}(i = 1,2,\\cdots,7263)\\) are non-negative integers. Find... | aime-beyond |
[
{
"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\nHow many squares at least should be marked on a \\(13\\times13\\) chessboard such that for any placement of a bishop o... | 24 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 101,
"raw_problem": "How many squares at least should be marked on a \\(13\\times13\\) chessboard such that for any placement of a bishop on the chessboard, the bishop can threaten at least one of the marked squares? (A bishop can... | aime-beyond |
[
{
"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 positive number \\(C\\)，such that for any six positive integers \\(x_1, x_2, \\ldots, x_6\\)，we have\... | 27 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 102,
"raw_problem": "Find the largest positive number \\(C\\)，such that for any six positive integers \\(x_1, x_2, \\ldots, x_6\\)，we have\\( \\left(x_1+\\cdots+x_6\\right)^3 \\geq C \\operatorname{gcd}\\left(x_1, x_2, \\ldots, x... | aime-beyond |
[
{
"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\nPlace 471 coins in the squares of an \\(n\\times n\\) board where each block can hold 1 coin, multiple coins, or no co... | 29 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 103,
"raw_problem": "Place 471 coins in the squares of an \\(n\\times n\\) board where each block can hold 1 coin, multiple coins, or no coin at all, such that the difference in the number of coins between any two adjacent squares... | aime-beyond |
[
{
"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\nFor integers \\(a_1,a_2,\\cdots,a_{7926}\\in\\{1,2,\\cdots,500\\}\\), denote \\(S = \\sum_{i = 1}^{7926}\\left[\\frac{... | 32 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 104,
"raw_problem": "For integers \\(a_1,a_2,\\cdots,a_{7926}\\in\\{1,2,\\cdots,500\\}\\), denote \\(S = \\sum_{i = 1}^{7926}\\left[\\frac{a_{i + 1}}{a_i}\\right]\\), where \\(a_{7927}=a_1\\). Find the minimal possible value of \\... | aime-beyond |
[
{
"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 maximum number of \"kings\" that can be placed on a \\(10\\times10\\) chessboard, such that each \"king\" can... | 40 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 105,
"raw_problem": "Find the maximum number of \"kings\" that can be placed on a \\(10\\times10\\) chessboard, such that each \"king\" can exactly attack ONE other \"king\" (a \"king\" can attack the chess piece in the square tha... | aime-beyond |
[
{
"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\nKevin throws three fair dice numbered from 1 to 6. Then he observes the results of the first throw of the three dice. ... | 42 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 106,
"raw_problem": "Kevin throws three fair dice numbered from 1 to 6. Then he observes the results of the first throw of the three dice. From these three dice, he can choose \\(n\\), where \\(0\\leq n\\leq3\\) dice to re-throw o... | aime-beyond |
[
{
"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 \\(m\\) be the maximal positive integer such that there exists complex numbers \\(r_1,\\cdots,r_{645}\\), which ar... | 46 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 107,
"raw_problem": "Let \\(m\\) be the maximal positive integer such that there exists complex numbers \\(r_1,\\cdots,r_{645}\\), which are not all zeros, satisfying \\(\\prod_{k = 1}^{645}(r_k + 1)=\\prod_{k = 1}^{645}(r_k^2+1)=... | aime-beyond |
[
{
"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\nGiven a \\(15\\times15\\) chessboard, there are \\(n\\) pieces placed on \\(n\\) individual squares respectively. For ... | 48 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 108,
"raw_problem": "Given a \\(15\\times15\\) chessboard, there are \\(n\\) pieces placed on \\(n\\) individual squares respectively. For each piece, a \\(5\\times5\\) square with its sides parallel to the boundaries of the chess... | aime-beyond |
[
{
"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 \\(\\left\\{a_n\\right\\}\\) satisfies \\(a_1=\\frac{2}{5}, a_n^2=a_{n+1}-2 a_n^2 a_{n+1}\\), find the residue of ... | 53 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 109,
"raw_problem": "Let \\(\\left\\{a_n\\right\\}\\) satisfies \\(a_1=\\frac{2}{5}, a_n^2=a_{n+1}-2 a_n^2 a_{n+1}\\), find the residue of \\(\\left[\\frac{1}{a_{2025}}\\right]\\) mod 407.",
"source_id": null,
"split": nul... | aime-beyond |
[
{
"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\nIt is known that one plane divides the space (three-dimensional) into two parts, two parallel planes divide the space ... | 64 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 110,
"raw_problem": "It is known that one plane divides the space (three-dimensional) into two parts, two parallel planes divide the space into three parts, and two intersecting planes divide the space into four parts; Consider th... | aime-beyond |
[
{
"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 \\(S = \\{\\frac{2y - 3}{2x - 1} \\mid \\min\\{\\vert x - 2\\vert,\\vert x + 2\\vert\\}+\\vert y\\vert\\leqslant1\... | 67 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 111,
"raw_problem": "Let \\(S = \\{\\frac{2y - 3}{2x - 1} \\mid \\min\\{\\vert x - 2\\vert,\\vert x + 2\\vert\\}+\\vert y\\vert\\leqslant1\\}\\), Assume the length of the set \\(S = \\frac{m}{p}\\) where \\(m\\) and \\(p\\) are co... | aime-beyond |
[
{
"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\nGiven that the quadrilateral inscribed in the sphere is an isosceles trapezoid \\(ABCD\\), where \\(AB = 2\\), \\(BC =... | 70 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 112,
"raw_problem": "Given that the quadrilateral inscribed in the sphere is an isosceles trapezoid \\(ABCD\\), where \\(AB = 2\\), \\(BC = AD = 3\\), \\(CD = 4\\), and the center of the sphere is \\(O\\). Points \\(E\\) and \\(F\... | aime-beyond |
[
{
"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\nSuppose there is a necklace composed of beads of two colors, red and blue, with a total of 16 beads. And the necklace ... | 88 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 113,
"raw_problem": "Suppose there is a necklace composed of beads of two colors, red and blue, with a total of 16 beads. And the necklace can be freely rotated and flipped (that is, the cases regarded as the same necklace need to... | aime-beyond |
[
{
"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 the plane be tiled with an infinite number of unit squares arranged in a checkerboard pattern. If another unit squ... | 90 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 114,
"raw_problem": "Let the plane be tiled with an infinite number of unit squares arranged in a checkerboard pattern. If another unit square falls randomly on this plane, and its position and orientation are independent of the t... | aime-beyond |
[
{
"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\nReal numbers \\(x_1 \\leq x_2 \\leq \\ldots \\leq x_9\\) such that \\(\\sum_{i=1}^9 x_i=1\\). Find the maximum value o... | 96 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 115,
"raw_problem": "Real numbers \\(x_1 \\leq x_2 \\leq \\ldots \\leq x_9\\) such that \\(\\sum_{i=1}^9 x_i=1\\). Find the maximum value of \n\\(\n\\frac{\\left(\\sum_{i=1}^9\\left|x_i-x_{5}\\right|\\right)^2 \\cdot \\sum_{1 \\le... | aime-beyond |
[
{
"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\nIn a square grid table composed of \\(16\\times16\\) unit squares with side length \\(1\\), \\(n\\) squares are select... | 108 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 116,
"raw_problem": "In a square grid table composed of \\(16\\times16\\) unit squares with side length \\(1\\), \\(n\\) squares are selected. In each of the selected squares, a directed diagonal is drawn. It is known that for any... | aime-beyond |
[
{
"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 set of 2025 squares in a \\(2025\\times2048\\) chessboard is called a generalized diagonal (simply called a diagonal... | 128 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 117,
"raw_problem": "A set of 2025 squares in a \\(2025\\times2048\\) chessboard is called a generalized diagonal (simply called a diagonal for short) if any two of these squares are neither in the same row nor in the same column.... | aime-beyond |
[
{
"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 \\(a_1,a_2,\\cdots\\) be an infinite sequence of positive integers with \\(a_1 = 1\\), such that\\(a_n\\mid a_k + ... | 143 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 118,
"raw_problem": "Let \\(a_1,a_2,\\cdots\\) be an infinite sequence of positive integers with \\(a_1 = 1\\), such that\\(a_n\\mid a_k + a_{k + 1}+\\cdots + a_{k + n - 1}\\) for all positive integers \\(k\\) and \\(n\\). Let \\... | aime-beyond |
[
{
"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\nIf integer \\(n\\) satisfies \\(a_1=n^2-10 n+23, a_2=n^2-9 n+97, a_3=n^2-12 n+46\\) are all primes, Find the sum of al... | 196 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 119,
"raw_problem": "If integer \\(n\\) satisfies \\(a_1=n^2-10 n+23, a_2=n^2-9 n+97, a_3=n^2-12 n+46\\) are all primes, Find the sum of all possible value of \\(a_1+a_2+a_3\\).",
"source_id": null,
"split": null,
"url... | aime-beyond |
[
{
"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\nThe 30 edges of a regular icosahedron are distinguished by labeling them 1, 2, ..., 30. How many different ways are th... | 224 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 120,
"raw_problem": "The 30 edges of a regular icosahedron are distinguished by labeling them 1, 2, ..., 30. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the... | aime-beyond |
[
{
"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\nAssume a positive integer \\(N\\) can be represented as summation of \\(k\\) consecutive positive integers, i.e. \\(N ... | 245 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 121,
"raw_problem": "Assume a positive integer \\(N\\) can be represented as summation of \\(k\\) consecutive positive integers, i.e. \\(N = a + (a + 1) + (a + 2) + \\cdots + (a + k - 1)\\), for \\(k = 1559\\), and can not be repr... | aime-beyond |
[
{
"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\nReal numbers \\(a,b,c>0\\) where \\(a+b+c=3\\), let the maximal value of \\((a^3+b^3+c^3)*(a+b)*(b+c)*(c+a)\\) be \\(p... | 247 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 122,
"raw_problem": "Real numbers \\(a,b,c>0\\) where \\(a+b+c=3\\), let the maximal value of \\((a^3+b^3+c^3)*(a+b)*(b+c)*(c+a)\\) be \\(p = \\frac{m}{n}\\) where \\(m\\) and \\(n\\) are coprimes. Find \\(m+n\\).",
"source_id... | aime-beyond |
[
{
"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\nTen points are randomly and independently generated with a uniform distribution inside a circle. Denote the probabilit... | 251 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 123,
"raw_problem": "Ten points are randomly and independently generated with a uniform distribution inside a circle. Denote the probability that the center of the circle lies outside the convex hull of these ten points as \\(p = ... | aime-beyond |
[
{
"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 coin is randomly drawn from a population. The probability that this coin shows heads when tossed once is \\(p\\), an... | 258 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 124,
"raw_problem": "A coin is randomly drawn from a population. The probability that this coin shows heads when tossed once is \\(p\\), and \\(p\\) follows a uniform distribution \\(U[0,1]\\). Subsequently, this randomly drawn co... | aime-beyond |
[
{
"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 is a function \\( g: \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N} \\) such that \\( g(0,0) = 0 \\), and for ... | 297 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 125,
"raw_problem": "There is a function \\( g: \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N} \\) such that \\( g(0,0) = 0 \\), and for any \\( x, y \\in \\mathbb{N} \\), there exists \\( n \\in \\mathbb{N} \\) such that\n\\[\\... | aime-beyond |
[
{
"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\nNine balls numbered \\(1, 2,\\cdots,9\\) are randomly placed on the nine equally-divided points of a circle, with one ... | 315 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 126,
"raw_problem": "Nine balls numbered \\(1, 2,\\cdots,9\\) are randomly placed on the nine equally-divided points of a circle, with one ball at each point. Let \\(S\\) be the sum of the absolute values of the differences betwee... | aime-beyond |
[
{
"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\nFor a positive integer \\(n\\), sort all the \\(l\\) positive factors of \\(n\\) as \\(d_1, d_2, \\cdots, d_l\\) from ... | 360 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 127,
"raw_problem": "For a positive integer \\(n\\), sort all the \\(l\\) positive factors of \\(n\\) as \\(d_1, d_2, \\cdots, d_l\\) from smallest to largest. If there exist positive integers \\(i, j \\in\\{1,2, \\cdots, l\\}\\),... | aime-beyond |
[
{
"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\n\\(m\\) distinct positive even numbers and \\(n\\) distinct positive odd numbers sum up to 2025. Find the maximal valu... | 384 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 128,
"raw_problem": "\\(m\\) distinct positive even numbers and \\(n\\) distinct positive odd numbers sum up to 2025. Find the maximal value of \\(5m+7n\\) among all such \\(m\\) and \\(n\\).",
"source_id": null,
"split": ... | aime-beyond |
[
{
"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\nBob has a convex 735-gon in the plane and\nwishes to construct the centroid of its vertices. He has no standard ruler ... | 386 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 129,
"raw_problem": "Bob has a convex 735-gon in the plane and\nwishes to construct the centroid of its vertices. He has no standard ruler or compass, but\nhe does have a device with which he can dissect the segment between two gi... | aime-beyond |
[
{
"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\nFor two sets \\(A\\) and \\(B\\), the set \\(A\\Delta B=\\{x|x\\in A\\text{ and }x\\notin B\\}\\cup\\{x|x\\in B\\text{... | 424 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 130,
"raw_problem": "For two sets \\(A\\) and \\(B\\), the set \\(A\\Delta B=\\{x|x\\in A\\text{ and }x\\notin B\\}\\cup\\{x|x\\in B\\text{ and }x\\notin A\\}\\) is defined. Let \\(A_1,A_2,\\cdots,A_{2^{312}}\\) be all the subsets... | aime-beyond |
[
{
"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\nReal numbers \\(x_1,x_2,\\cdots,x_n\\) satisfy \\(|x_1|+|x_2|+\\cdots+|x_{3526}| = 1\\). Let \\(M\\) be the maximal po... | 432 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 131,
"raw_problem": "Real numbers \\(x_1,x_2,\\cdots,x_n\\) satisfy \\(|x_1|+|x_2|+\\cdots+|x_{3526}| = 1\\). Let \\(M\\) be the maximal possible value of \\(\\frac{1}{|x_1|+|x_1 - x_2|+|x_1 + x_2 - x_3|+\\cdots+|x_1 + x_2+\\cdots... | aime-beyond |
[
{
"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\nGiven \\(x_1^2+y_1^2+z_1^2=1,6 x_2+3 y_2+2 z_2=12\\), where \\(x_1, y_1, z_1 \\in \\mathbb{R}, x_2, y_2, z_2 \\in(0,+\... | 454 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 132,
"raw_problem": "Given \\(x_1^2+y_1^2+z_1^2=1,6 x_2+3 y_2+2 z_2=12\\), where \\(x_1, y_1, z_1 \\in \\mathbb{R}, x_2, y_2, z_2 \\in(0,+\\infty)\\). Let the minimal value of \\(\\max \\left\\{\\left|x_2-x_1\\right|,\\left|y_2-y_... | aime-beyond |
[
{
"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 \\(12\\times12\\) matrix \\(A\\) satisfies that each row has exactly \\(6\\) elements equal to \\(1\\) and \\(6\\) e... | 462 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 133,
"raw_problem": "A \\(12\\times12\\) matrix \\(A\\) satisfies that each row has exactly \\(6\\) elements equal to \\(1\\) and \\(6\\) elements equal to \\(0\\). Then, how many matrices \\(A\\) are there such that \\(A^{2}+6A\\... | aime-beyond |
[
{
"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 \\(\\mathbb{F}_{2027}\\) be the field of residues modulo \\(2027\\). Let \\(W\\) be the smallest set of polynomial... | 505 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 134,
"raw_problem": "Let \\(\\mathbb{F}_{2027}\\) be the field of residues modulo \\(2027\\). Let \\(W\\) be the smallest set of polynomials with coefficients in \\(\\mathbb{F}_{2027}\\) such that the polynomials \\(x+1\\) and \\(... | aime-beyond |
[
{
"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\nFor every real number \\(x\\), let \\(f(x)=\\sum_{n\\in S_x}\\frac{1}{2^n}\\), where \\(S_x\\) is the set of positive ... | 571 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 135,
"raw_problem": "For every real number \\(x\\), let \\(f(x)=\\sum_{n\\in S_x}\\frac{1}{2^n}\\), where \\(S_x\\) is the set of positive integers \\(n\\) such that\\(\\lfloor nx\\rfloor\\) is even. Let\\(L_{max}\\) be the larges... | aime-beyond |
[
{
"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 a pair of integers \\(a, b\\) such as \\(a^2+b^2 \\leq 19\\), and there exists real numbers \\(x, y\\) such t... | 576 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 136,
"raw_problem": "Consider a pair of integers \\(a, b\\) such as \\(a^2+b^2 \\leq 19\\), and there exists real numbers \\(x, y\\) such that \\(\\frac{a}{\\sqrt{x}}+\\frac{2}{y+3 x}=\\frac{b}{\\sqrt{y}}-\\frac{2}{y+3 x}=1\\)．Fin... | aime-beyond |
[
{
"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 minimal number \\(C\\) satisfying：\n\\[\n\\sum_{0 \\leq i,j \\leq n} 4^{i+j+1} C_{i+j}^i x_i x_j+\\sum_{0 \\l... | 592 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 137,
"raw_problem": "Find the minimal number \\(C\\) satisfying：\n\\[\n\\sum_{0 \\leq i,j \\leq n} 4^{i+j+1} C_{i+j}^i x_i x_j+\\sum_{0 \\leq i \\leq n} x_i \\geq -\\frac{1}{4144}C \n\\]\nfor all positive integers \\(n\\) and \\(x... | aime-beyond |
[
{
"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 positive integers \\(a_1<a_2<\\cdots<a_{38}\\) satisfying \\(a_1+a_2+\\cdots+a_{38}=1997\\). Let \\(b_k=\\min... | 666 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 138,
"raw_problem": "Consider positive integers \\(a_1<a_2<\\cdots<a_{38}\\) satisfying \\(a_1+a_2+\\cdots+a_{38}=1997\\). Let \\(b_k=\\min \\left\\{a_1+a_2+\\cdots+a_k, a_{k+1}+a_{k+2}+\\cdots+a_{38}\\right\\}\\). Let \\(S=\\max ... | aime-beyond |
[
{
"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\nArrange the numbers \\(1, 2,\\dots,333^{2}\\) into a \\(333\\times333\\) number table, with each number appearing exac... | 667 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 139,
"raw_problem": "Arrange the numbers \\(1, 2,\\dots,333^{2}\\) into a \\(333\\times333\\) number table, with each number appearing exactly once. Calculate the ratio of the larger number to the smaller number for every pair of ... | aime-beyond |
[
{
"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 the radius of the circle \\(O\\) be \\(r = 1\\), and \\(M\\) be a point on the edges of regular dodecagon (12-side... | 670 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 140,
"raw_problem": "Let the radius of the circle \\(O\\) be \\(r = 1\\), and \\(M\\) be a point on the edges of regular dodecagon (12-sided polygon) inscribed in the circle \\(O\\). \\(E\\) and \\(F\\) are a pair of diametrically... | aime-beyond |
[
{
"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\nFor a finite non-empty set \\(X\\) of real numbers, denote the number of elements in \\(X\\) as \\(\\vert X\\vert\\), ... | 724 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 141,
"raw_problem": "For a finite non-empty set \\(X\\) of real numbers, denote the number of elements in \\(X\\) as \\(\\vert X\\vert\\), and \\(f(X)=\\frac{1}{\\vert X\\vert}(\\sum_{a\\in X}a)\\). The pair of sets \\((A,B)\\) sa... | aime-beyond |
[
{
"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 is a sphere with radius \\(\\sqrt{6}/36\\) moving freely inside a tetrahedron with side length 1. Let \\(V\\) be... | 820 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 142,
"raw_problem": "There is a sphere with radius \\(\\sqrt{6}/36\\) moving freely inside a tetrahedron with side length 1. Let \\(V\\) be the volume of the reachable regions of the center of the sphere. Find \\(\\lfloor 1/V^2 \... | aime-beyond |
[
{
"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 minimal positive integer \\(n \\geqslant 866\\) satisfying: there exists a bijection \\(f\\) from \\(\\{1,\\c... | 876 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 143,
"raw_problem": "Find the minimal positive integer \\(n \\geqslant 866\\) satisfying: there exists a bijection \\(f\\) from \\(\\{1,\\cdots,n\\}\\) to itself, such that for any \\(x,y\\in\\{1,\\cdots,n\\}\\), if \\(n\\) divide... | aime-beyond |
[
{
"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_i=\\frac{i}{1771}(i=0,1, \\cdots, 1771)\\), \\(S=\\sum_{i=0}^{1771} \\frac{x_i^5}{5 x_i^4-10 x_i^3+10 x_i^2-5... | 886 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 144,
"raw_problem": "Let \\(x_i=\\frac{i}{1771}(i=0,1, \\cdots, 1771)\\), \\(S=\\sum_{i=0}^{1771} \\frac{x_i^5}{5 x_i^4-10 x_i^3+10 x_i^2-5 x_i+1}\\), Find \\(S\\).",
"source_id": null,
"split": null,
"url": null
}
} | aime-beyond |
[
{
"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\nGiven non-negative real numbers \\(x, y, z\\) satisfying \\(\\sqrt{x^2+y^2}+z=1\\), for positive real number \\(k\\), ... | 979 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 145,
"raw_problem": "Given non-negative real numbers \\(x, y, z\\) satisfying \\(\\sqrt{x^2+y^2}+z=1\\), for positive real number \\(k\\), define \\(f(k)\\) as the maximal value of \\(4(xy+kxz)\\). Find the largest integer no more... | aime-beyond |
[
{
"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 \\(I_{n}=\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\sin^{2}(nt)}{\\sin t}dt\\), find the largest integer smaller than \\(... | 1151 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 146,
"raw_problem": "Let \\(I_{n}=\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\sin^{2}(nt)}{\\sin t}dt\\), find the largest integer smaller than \\(1000 \\times \\lim_{n\\rightarrow\\infty}\\frac{I_{n}}{\\log_10{n}}\\).",
"source_id": ... | aime-beyond |
[
{
"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 minimal constant \\(k\\), such that for any positive real numbers \\(x,y,z\\), the inequality \\(\\frac{x}{\\... | 1250 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 147,
"raw_problem": "Find the minimal constant \\(k\\), such that for any positive real numbers \\(x,y,z\\), the inequality \\(\\frac{x}{\\sqrt{x+y}}+\\frac{y}{\\sqrt{y+z}}+\\frac{z}{\\sqrt{z+x}}\\leq \\frac{k}{1000}\\sqrt{x+y+z}\... | aime-beyond |
[
{
"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\nGiven 1893 people and 66 kinds of shapes. Each person holds one cookie of each of the 66 shapes, and the total mass of... | 1403 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 148,
"raw_problem": "Given 1893 people and 66 kinds of shapes. Each person holds one cookie of each of the 66 shapes, and the total mass of these 66 cookies is 1 (the mass of each cookie is a positive real number; the masses of co... | aime-beyond |
[
{
"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 \\(S = \\{1,2,\\cdots,6172\\}\\) and \\(M\\) is the number of functions \\(f:S\\rightarrow S\\) that satisfy the f... | 1543 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 149,
"raw_problem": "Let \\(S = \\{1,2,\\cdots,6172\\}\\) and \\(M\\) is the number of functions \\(f:S\\rightarrow S\\) that satisfy the following conditions: \n(1) \\(f\\) is a one-to-one mapping. \n(2) For any positive integer ... | aime-beyond |
[
{
"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 smallest positive integer \\(t\\), such that for any real numbers \\(a_1, a_2, \\cdots, a_{4467}\\), the ineq... | 2234 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 150,
"raw_problem": "Find the smallest positive integer \\(t\\), such that for any real numbers \\(a_1, a_2, \\cdots, a_{4467}\\), the inequality \\(\\max(|a_1|, |a_2|, \\cdots, |a_{4467}|) \\leq t \\cdot \\max(|a_1 + a_2|, |a_2 +... | aime-beyond |
[
{
"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\n\\(x_1,x_2,\\dots,x_{2025}\\) are positive real numbers. Let \\(M\\) be the minimal value of \\(\\max\\{x_1,\\frac{1}{... | 2304 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 151,
"raw_problem": "\\(x_1,x_2,\\dots,x_{2025}\\) are positive real numbers. Let \\(M\\) be the minimal value of \\(\\max\\{x_1,\\frac{1}{x_1}+x_2,\\dots,\\frac{1}{x_{2024}}+x_{2025},\\frac{1}{x_{2025}}+1\\}\\). Suppose the a pol... | aime-beyond |
[
{
"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\nIn a \\(60\\times60\\) square grid of cells, the integers \\(1, 2,\\cdots,3600\\) are filled in such a way that the su... | 3571 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 152,
"raw_problem": "In a \\(60\\times60\\) square grid of cells, the integers \\(1, 2,\\cdots,3600\\) are filled in such a way that the sum of the numbers in every two adjacent cells (cells that share a common side are called adj... | aime-beyond |
[
{
"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 non-real complex number \\(z\\) satisfying \\(z^{4051}+z^{4050}+5=0\\). Find the number of such \\(z\\)'s (multipl... | 4050 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 153,
"raw_problem": "Let non-real complex number \\(z\\) satisfying \\(z^{4051}+z^{4050}+5=0\\). Find the number of such \\(z\\)'s (multiplicity are counted)．",
"source_id": null,
"split": null,
"url": null
}
} | aime-beyond |
[
{
"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(x),Q(x)\\) be distinct 4046-degree real polynomials with non-zero coefficients. Let \\(r\\) be the number of ... | 6068 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 154,
"raw_problem": "Let \\(P(x),Q(x)\\) be distinct 4046-degree real polynomials with non-zero coefficients. Let \\(r\\) be the number of their common real roots (multiplicity counted) and \\(s\\) be the number of their common te... | aime-beyond |
[
{
"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 regular octagon is divided into 8 regions, namely \\(A\\), \\(B\\), \\(C\\), \\(D\\), \\(E\\), \\(F\\), \\(G\\), and... | 6564 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 155,
"raw_problem": "A regular octagon is divided into 8 regions, namely \\(A\\), \\(B\\), \\(C\\), \\(D\\), \\(E\\), \\(F\\), \\(G\\), and \\(H\\) by connecting the center point with each vertex. Now we place one ball in each reg... | aime-beyond |
[
{
"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 100 observers inside a finite circular playground (the positions of any two observers are different). Each o... | 8700 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 156,
"raw_problem": "There are 100 observers inside a finite circular playground (the positions of any two observers are different). Each observer has a viewing angle of \\(100^{\\circ}\\) and can observe all the other observers w... | aime-beyond |
[
{
"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\nDenote \\(x\\succ y\\) as \\(0\\lt x - y\\leqslant9\\) or \\(y - x\\gt9\\). Let \\(a,b,c,d\\in\\{1,2,\\cdots,19\\}\\).... | 9120 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 157,
"raw_problem": "Denote \\(x\\succ y\\) as \\(0\\lt x - y\\leqslant9\\) or \\(y - x\\gt9\\). Let \\(a,b,c,d\\in\\{1,2,\\cdots,19\\}\\). For the cases where \\(a\\succ b\\), \\(b\\succ c\\), \\(c\\succ d\\) and \\(d\\succ a\\),... | aime-beyond |
[
{
"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_k\\in [-2,2],k=1,2\\dots,4698\\) satisfying \\(x_1+x_2+\\dots +x_{4698}=0\\). Find the maximal value of \\(M=... | 9396 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 158,
"raw_problem": "Let \\(x_k\\in [-2,2],k=1,2\\dots,4698\\) satisfying \\(x_1+x_2+\\dots +x_{4698}=0\\). Find the maximal value of \\(M=x_1^3+x_2^3+\\dots +x_{4698}^3\\).",
"source_id": null,
"split": null,
"url": n... | aime-beyond |
[
{
"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\nThe rectangle \\(R\\) is divided into 2024 small rectangles, and the sides of each small rectangle are parallel to the... | 10211 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 159,
"raw_problem": "The rectangle \\(R\\) is divided into 2024 small rectangles, and the sides of each small rectangle are parallel to the sides of \\(R\\). The vertices of the small rectangles are called nodes. A line segment on... | aime-beyond |
[
{
"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 \\(G\\) be a 160-order simple graph. For any vertex \\(u\\), there exists another vertex \\(v\\), such that \\(u\\... | 10320 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 160,
"raw_problem": "Let \\(G\\) be a 160-order simple graph. For any vertex \\(u\\), there exists another vertex \\(v\\), such that \\(u\\) and \\(v\\) are adjacent, and there exists no vertices adjacent to both \\(u\\) and \\(v\... | aime-beyond |
[
{
"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\nChoose a subset of integers from 1 to 15 such that no three consecutive integers are in the chosen set (you can choose... | 10609 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 161,
"raw_problem": "Choose a subset of integers from 1 to 15 such that no three consecutive integers are in the chosen set (you can choose an empty set). How many different sets are there?",
"source_id": null,
"split": nu... | aime-beyond |
[
{
"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 \\(\\mathbb{Z} / n\\mathbb{Z}\\) be the set of all integers modulo \\(n\\) (hence \\(\\mathbb{Z} / n\\mathbb{Z}\\)... | 11021 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 162,
"raw_problem": "Let \\(\\mathbb{Z} / n\\mathbb{Z}\\) be the set of all integers modulo \\(n\\) (hence \\(\\mathbb{Z} / n\\mathbb{Z}\\) contains \\(n\\) elements). Find the second smallest composite number \\(n\\), such that t... | aime-beyond |
[
{
"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\nThe Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera\nhas a row of 7873 such coins altern... | 15744 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 163,
"raw_problem": "The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera\nhas a row of 7873 such coins alternately tails-up and heads-up, with the leftmost\ncoin tails-up.\nIn a move, Vera may flip ov... | aime-beyond |
[
{
"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 all the lattice points \\((x, y)\\) where \\(1\\leqslant x\\leqslant9973\\) and \\(1\\leqslant y\\leqslant9973\\... | 24931 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 164,
"raw_problem": "Among all the lattice points \\((x, y)\\) where \\(1\\leqslant x\\leqslant9973\\) and \\(1\\leqslant y\\leqslant9973\\), some lattice points are to be selected. The restrictive condition is that among the sele... | aime-beyond |
[
{
"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 edge of the complete graph \\(K_{9097}\\) is labeled with \\(1\\), \\(2\\), or \\(3\\) such that the sum of the l... | 27290 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 165,
"raw_problem": "Each edge of the complete graph \\(K_{9097}\\) is labeled with \\(1\\), \\(2\\), or \\(3\\) such that the sum of the labels of any triangle (formed by three edges) is at least \\(5\\). Given that the minimum p... | aime-beyond |
[
{
"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\nIntegers \\(a_1,a_2,\\cdots,a_{2025}\\) satisfy \\(1 = a_1\\leq a_2\\leq\\cdots\\leq a_{2025}=199\\). Let \\(f=(a_1^2 ... | 29800 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 166,
"raw_problem": "Integers \\(a_1,a_2,\\cdots,a_{2025}\\) satisfy \\(1 = a_1\\leq a_2\\leq\\cdots\\leq a_{2025}=199\\). Let \\(f=(a_1^2 + a_2^2 + \\cdots + a_{2025}^2)-(a_1a_3 + a_2a_4 + a_3a_5 + \\cdots + a_{2023}a_{2025})\\).... | aime-beyond |
[
{
"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 \\(S\\) be the set of integers \\(s\\) satisfying \\(s\\geq3456\\), and there exists positive integers \\(a\\), \\... | 34628 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 167,
"raw_problem": "Let \\(S\\) be the set of integers \\(s\\) satisfying \\(s\\geq3456\\), and there exists positive integers \\(a\\), \\(b\\), \\(c\\), \\(d\\) such that \\(s = a + b + c + d\\) and \\(s\\) divides \\(abc + abd ... | aime-beyond |
[
{
"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\nThe number of ways to color an \\(n \\times n\\) chessboard with black and white such that no neighboring two blocks c... | 56751 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 168,
"raw_problem": "The number of ways to color an \\(n \\times n\\) chessboard with black and white such that no neighboring two blocks can be black simultaneously is \\(x_n\\). Find \\(\\sum_{n=2}^5 x_n\\)．",
"source_id": n... | aime-beyond |
[
{
"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\nThe values of the 20 numbers \\(a_1, a_2, \\cdots, a_{20}\\) can be either \\(0\\) or \\(1\\). If two adjacent terms \... | 116280 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 169,
"raw_problem": "The values of the 20 numbers \\(a_1, a_2, \\cdots, a_{20}\\) can be either \\(0\\) or \\(1\\). If two adjacent terms \\(a_k\\) and \\(a_{k + 1}\\) are \\(1\\) and \\(0\\) respectively, then it is called a \"re... | aime-beyond |
[
{
"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\nReal numbers \\(x_{1},x_{2},\\cdots,x_{1997}\\) satisfy the following two conditions: (1)\\(-\\frac{1}{\\sqrt{3}}\\leq... | 189548 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 170,
"raw_problem": "Real numbers \\(x_{1},x_{2},\\cdots,x_{1997}\\) satisfy the following two conditions: (1)\\(-\\frac{1}{\\sqrt{3}}\\leq x_{i}\\leq\\sqrt{3}\\)（\\(i = 1,2,\\cdots,1997\\); (2)\\(x_{1}+x_{2}+\\cdots +x_{1997}=-31... | aime-beyond |
[
{
"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 \\(a_1,a_2,...,a_m\\) be \\(m\\) distince positive integers such that \\(a_1\\cdot a_2\\cdot...\\cdot a_m\\) has 6... | 367236 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 171,
"raw_problem": "Let \\(a_1,a_2,...,a_m\\) be \\(m\\) distince positive integers such that \\(a_1\\cdot a_2\\cdot...\\cdot a_m\\) has 606 prime factors. If the product of any number of terms among \\(a_1, a_2, \\cdots, a_m\\) ... | aime-beyond |
[
{
"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 1038 equally spaced points on a circular track \\(\\gamma\\) of circumference 1038. The points are labeled \... | 537686 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 172,
"raw_problem": "There are 1038 equally spaced points on a circular track \\(\\gamma\\) of circumference 1038. The points are labeled \\(A_1, A_2, \\dots, A_{1038}\\) in some order, each label used once. Initially, Bunbun the ... | aime-beyond |
[
{
"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 pairs \\((f, g)\\) of functions from the set of nonnegative integers to itself such that \\(f(0) \\ge f(1) \\... | 794430 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 173,
"raw_problem": "Consider pairs \\((f, g)\\) of functions from the set of nonnegative integers to itself such that \\(f(0) \\ge f(1) \\geq f(2) \\geq \\cdots \\ge f(2016) \\ge 0\\) and \\(f(0)+f(1)+f(2)+\\cdots +f(2016) \\le 2... | aime-beyond |
[
{
"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 \\(G\\) be a \\(499\\times3997\\) grid table, and a number is filled in each cell of \\(G\\). A rectangular sub-gr... | 999499 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 174,
"raw_problem": "Let \\(G\\) be a \\(499\\times3997\\) grid table, and a number is filled in each cell of \\(G\\). A rectangular sub-grid table \\(S\\) of \\(G\\) is called \"outstanding\" if the number in each cell of \\(S\\)... | aime-beyond |
[
{
"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\nGiven positive integer \\(n \\geq 3\\), \\(x_1, x_2, \\cdots, x_n \\in[0,1]\\), consider the set \\(A=\\left\\{(i, j)\... | 1026167 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 175,
"raw_problem": "Given positive integer \\(n \\geq 3\\), \\(x_1, x_2, \\cdots, x_n \\in[0,1]\\), consider the set \\(A=\\left\\{(i, j)\\left|1 \\leq i<j \\leq n, i, j \\in \\mathbb{N}^*, i\\right| j\\right\\}\\), let \\(d(n)\\... | aime-beyond |
[
{
"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 a regular 25-prism. By connecting the face diagonals of each lateral face, 25 intersection points are generat... | 1342182 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 176,
"raw_problem": "Consider a regular 25-prism. By connecting the face diagonals of each lateral face, 25 intersection points are generated, and then the lateral edges are removed. Now, we perform a three-coloring on these 25 in... | aime-beyond |
[
{
"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\nIf there exist positive integers \\(a\\) and \\(b\\) (\\(a < b\\)) such that \\(a^{3}=(ka + 1)b + k\\), then the integ... | 1995002 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 177,
"raw_problem": "If there exist positive integers \\(a\\) and \\(b\\) (\\(a < b\\)) such that \\(a^{3}=(ka + 1)b + k\\), then the integer \\(k\\) is called \"good\". Then, find the sum of all \"good\" integers \\(k\\) that do ... | aime-beyond |
[
{
"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\nKumar has a regular 2024-gon, and there is a coin on each vertex of it. In one operation, he selects a vertex, takes o... | 2022676 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 178,
"raw_problem": "Kumar has a regular 2024-gon, and there is a coin on each vertex of it. In one operation, he selects a vertex, takes one coin from each of the two adjacent vertices to this selected vertex, adds one coin to th... | aime-beyond |
[
{
"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\nOn a \\(2025\\times2025\\) grid table, one white piece and one black piece are randomly placed in different cells. Ali... | 2050312 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 179,
"raw_problem": "On a \\(2025\\times2025\\) grid table, one white piece and one black piece are randomly placed in different cells. Alice moves the white piece first, then Bob moves the black piece, and they move alternately, ... | aime-beyond |
[
{
"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\nThe side lengths of a convex 2025-gon are \\(x_1, x_2, \\cdots, x_n\\), where \\(x_i\\in\\mathbb{Z}_{+}\\) (\\(1\\leq ... | 2061442 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 180,
"raw_problem": "The side lengths of a convex 2025-gon are \\(x_1, x_2, \\cdots, x_n\\), where \\(x_i\\in\\mathbb{Z}_{+}\\) (\\(1\\leq i\\leq n\\)). Let \\(f(m)\\) denote the minimum value of \\(x_1x_2\\cdots x_n\\) when the p... | aime-beyond |
[
{
"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\nAt a party, the host wants to distribute 2025 pieces of cookies of the same size to \\(n\\) guests and follow the foll... | 2085451 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 181,
"raw_problem": "At a party, the host wants to distribute 2025 pieces of cookies of the same size to \\(n\\) guests and follow the following two rules: (1) Each cookie can be divided into at most two parts (does not have to be... | aime-beyond |
[
{
"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\nIn an \\(n\\times n\\) (\\(n\\in\\mathbb{N}^*, n\\geq2\\)) chessboard, each unit square is filled with the number \\(0... | 3077492 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 182,
"raw_problem": "In an \\(n\\times n\\) (\\(n\\in\\mathbb{N}^*, n\\geq2\\)) chessboard, each unit square is filled with the number \\(0\\). In each operation, two unit squares with a common side are selected, and the numbers i... | aime-beyond |
[
{
"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\nFor positive integer \\(n\\) and real numbers \\(a_1, a_2, \\ldots, a_n \\in[-1,1]\\). Let \\(S=\\sum_{i=1}^n a_i\\) a... | 3847537 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 183,
"raw_problem": "For positive integer \\(n\\) and real numbers \\(a_1, a_2, \\ldots, a_n \\in[-1,1]\\). Let \\(S=\\sum_{i=1}^n a_i\\) and \\(f(n)\\) be the maximum value of \\(\\sum_{i=1}^n\\left|\\frac{S-a_i}{n}-a_i\\right|\\... | aime-beyond |
[
{
"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\nNon negative real numbers \\(x_1, x_2, \\cdots, x_{2025}\\) sum up to 1. Let \\(S = \\frac{m}{n}\\) be the maximal val... | 4104674 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 184,
"raw_problem": "Non negative real numbers \\(x_1, x_2, \\cdots, x_{2025}\\) sum up to 1. Let \\(S = \\frac{m}{n}\\) be the maximal value of \\(\\sum_{1 \\leq i, j \\leq 2025} \\frac{2 i-1}{2 j-1} x_i x_j\\) where \\(m\\) and ... | aime-beyond |
[
{
"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 13 cards, on which the positive integers \\(2,3,\\cdots,14\\) are written respectively. These cards are rand... | 6081079 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 185,
"raw_problem": "There are 13 cards, on which the positive integers \\(2,3,\\cdots,14\\) are written respectively. These cards are randomly arranged in a row from left to right, and the number on the \\(i\\)-th card is denoted... | aime-beyond |
[
{
"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 \\(A = \\{1,2,\\cdots,12\\}\\), and \\(f\\) be a one-to-one mapping from \\(A\\) onto \\(A\\). Denote \\(f_1(x)=f(... | 13685760 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 186,
"raw_problem": "Let \\(A = \\{1,2,\\cdots,12\\}\\), and \\(f\\) be a one-to-one mapping from \\(A\\) onto \\(A\\). Denote \\(f_1(x)=f(x)\\) and \\(f_{n + 1}(x)=f(f_n(x))\\). The mapping \\(f\\) satisfies the following conditi... | aime-beyond |
[
{
"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 7671 piles of coins, denoted as \\(S_1, S_2, \\cdots, S_{7671}\\) respectively. Initially, for each \\(k = 1... | 14711060 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 187,
"raw_problem": "There are 7671 piles of coins, denoted as \\(S_1, S_2, \\cdots, S_{7671}\\) respectively. Initially, for each \\(k = 1, 2, \\cdots, 7671\\), there are \\(k\\) coins in the pile \\(S_k\\). In one operation, an ... | aime-beyond |
[
{
"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\nTake out 17 balls from an ample supply of red, green, and blue balls (these balls are completely identical except for ... | 28357376 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 188,
"raw_problem": "Take out 17 balls from an ample supply of red, green, and blue balls (these balls are completely identical except for their colors) and arrange them in a row. It is required that no two green balls are adjacen... | aime-beyond |
[
{
"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 a graph with 199 vertices, where each pair of vertices are connected by an edge. 199 of the edges are two-way... | 33124147 | {
"ability": "MATH",
"data_source": "aime-beyond",
"extra_info": {
"index": 189,
"raw_problem": "Consider a graph with 199 vertices, where each pair of vertices are connected by an edge. 199 of the edges are two-way and the rest of the edges are one-way. We called a quadruple of vertices \\(A, B, C, D\\) ... | aime-beyond |
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