prompt listlengths 1 1 | label stringlengths 1 48 | metadata dict |
|---|---|---|
[
{
"role": "user",
"content": "In the triangle $ABC$, we have $\\angle ABC = 100^\\circ$, $\\angle ACB = 65^\\circ$, $M \\in AB$, $N \\in AC$, and $\\angle MCB = 55^\\circ$, $\\angle NBC = 80^\\circ$. Find $\\angle NMC$. Please put your final answer in \\boxed{}."
}
] | 25 | {
"ground_truth": "25",
"data_type": "klear_math",
"num_tokens": 89
} |
[
{
"role": "user",
"content": "A regular $2012$-gon is inscribed in a circle. Find the maximal $k$ such that we can choose $k$ vertices from given $2012$ and construct a convex $k$-gon without parallel sides.\n\nPlease put your final answer in \\boxed{}."
}
] | 1509 | {
"ground_truth": "1509",
"data_type": "klear_math",
"num_tokens": 73
} |
[
{
"role": "user",
"content": "Example 3 Find the minimum value of the function $u(x, y)=x^{2}+\\frac{81}{x^{2}}-2 x y+\\frac{18}{x} \\sqrt{2-y^{2}}$ with real numbers $x, y$ as variables.\n(1991 \"Hope Cup\" Invitational Competition Question)\n\nPlease put your final answer in \\boxed{}."
}
] | 6 | {
"ground_truth": "6",
"data_type": "klear_math",
"num_tokens": 97
} |
[
{
"role": "user",
"content": "4. On graph paper, a right-angled triangle with legs equal to 7 cells was drawn (see fig.). Then all the grid lines inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing?\n\n#\n\nPlease put your final answer in \\boxed{}."
... | 28 | {
"ground_truth": "28",
"data_type": "klear_math",
"num_tokens": 72
} |
[
{
"role": "user",
"content": "Example 3 㷵 On a board, there is a convex 2011-gon, and Betya draws its diagonals one by one. It is known that each diagonal drawn intersects at most one of the previously drawn diagonals at an interior point. Question: What is the maximum number of diagonals Betya can draw? [3... | 4016 | {
"ground_truth": "4016",
"data_type": "klear_math",
"num_tokens": 96
} |
[
{
"role": "user",
"content": "At a party, there were some couples attending. As they arrive, each person gets to talk with all the other persons who are already in the room. During the party, after all the guests arrive, groups of persons form such that:\n1. No two persons forming a couple belong to the sam... | 4 | {
"ground_truth": "4",
"data_type": "klear_math",
"num_tokens": 131
} |
[
{
"role": "user",
"content": "Find the sum of the decimal digits of the number\n$$\\sum_{k=1}^{99} 5k(k + 1)(k^2 + k + 1).$$\n Please put your final answer in \\boxed{}."
}
] | 72 | {
"ground_truth": "72",
"data_type": "klear_math",
"num_tokens": 61
} |
[
{
"role": "user",
"content": "Leticia has a $9 \\times 9$ board. She says that two squares are \"friends\" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of... | 486 | {
"ground_truth": "486",
"data_type": "klear_math",
"num_tokens": 218
} |
[
{
"role": "user",
"content": "In the game of *Winners Make Zeros*, a pair of positive integers \\((m,n)\\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:\n\n- If \\(m \\geq n\\), the player chooses a positive integer \\(c\\) such that \\(m-cn \\geq 0\\),... | 999 | {
"ground_truth": "999",
"data_type": "klear_math",
"num_tokens": 250
} |
[
{
"role": "user",
"content": "Let $f:\\mathbb{N} \\longrightarrow \\mathbb{N}$ be such that for every positive integer $n$ , followings are satisfied.\ni. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$ .\n(Here, $\\mathbb{N}$ is the set of all positive integers.)\n\nPlease put your... | 4026 | {
"ground_truth": "4026",
"data_type": "klear_math",
"num_tokens": 119
} |
[
{
"role": "user",
"content": "Define a crossword puzzle to be a 15 × 15 grid of squares, each of which is either black or white. In\na crossword puzzle, define a word to be a sequence of one or more consecutive white squares in a row\nor column such that the squares immediately before and after the sequence... | 4900 | {
"ground_truth": "4900",
"data_type": "klear_math",
"num_tokens": 162
} |
[
{
"role": "user",
"content": "A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_n\\le.4$ for all $n$ such that $1\\le n\\le9$ is give... | 660 | {
"ground_truth": "660",
"data_type": "klear_math",
"num_tokens": 173
} |
[
{
"role": "user",
"content": "At the base of the pyramid \\(S A_{1} A_{2} \\ldots A_{n}\\) lies point \\(O\\), and it is given that \\(S A_{1}=S A_{2}=\\ldots=S A_{n}\\) and \\(\\angle S A_{1} O = \\angle S A_{2} O = \\ldots = \\angle S A_{n} O\\).\n\nFor what smallest value of \\(n\\) does it follow that \... | 5 | {
"ground_truth": "5",
"data_type": "klear_math",
"num_tokens": 132
} |
[
{
"role": "user",
"content": "A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$? Please put your final answer in \\boxed{}."
}
] | 250 | {
"ground_truth": "250",
"data_type": "klear_math",
"num_tokens": 79
} |
[
{
"role": "user",
"content": "Example 8 Let $S=\\{1,2, \\cdots, 98\\}$. Find the smallest natural number $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always one group in which there is 1 number tha... | 50 | {
"ground_truth": "50",
"data_type": "klear_math",
"num_tokens": 135
} |
[
{
"role": "user",
"content": "Given that \\( m \\) and \\( n \\) are two distinct positive integers and the last four digits of \\( 2019^{m} \\) and \\( 2019^{n} \\) are the same, find the minimum value of \\( m+n \\).\n\nPlease put your final answer in \\boxed{}."
}
] | 502 | {
"ground_truth": "502",
"data_type": "klear_math",
"num_tokens": 79
} |
[
{
"role": "user",
"content": "Determine the abscissa of convergence, \\( \\sigma_c \\), for the Dirichlet series \\( \\sum_{n=1}^{\\infty} \\frac{n^{i \\alpha n}}{n^s} \\), where \\( \\alpha \\in \\mathbb{Z} \\).\n\nPlease put your final answer in \\boxed{}."
}
] | 0 | {
"ground_truth": "0",
"data_type": "klear_math",
"num_tokens": 84
} |
[
{
"role": "user",
"content": "Evaluate the expression \\(A + B + C + AB + BC + AC + ABC\\) where:\n\\[ A = \\frac{2\\sqrt{3}}{3}i \\cdot \\cos\\left(\\frac{\\pi}{6} + \\frac{i}{3}\\text{arcsinh}\\left(\\frac{3\\sqrt{3}}{2}\\right)\\right) \\]\n\\[ B = \\frac{2\\sqrt{3}}{3}i \\cdot \\cos\\left(\\frac{5\\pi}{... | 0 | {
"ground_truth": "0",
"data_type": "klear_math",
"num_tokens": 230
} |
[
{
"role": "user",
"content": "let N be the smallest 5-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 10000. Find Q + R. Please put your final answer ... | 3548 | {
"ground_truth": "3548",
"data_type": "klear_math",
"num_tokens": 82
} |
[
{
"role": "user",
"content": "Let $\\triangle ABC$ have median $CM$ ( $M\\in AB$ ) and circumcenter $O$ . The circumcircle of $\\triangle AMO$ bisects $CM$ . Determine the least possible perimeter of $\\triangle ABC$ if it has integer side lengths.\n\nPlease put your final answer in \\boxed{}."
... | 24 | {
"ground_truth": "24",
"data_type": "klear_math",
"num_tokens": 87
} |
[
{
"role": "user",
"content": "Determine the number of positive integer solutions to the equation \\( \\phi(x) = 2^{2013} \\), where \\( \\phi \\) is the Euler-totient function.\n\nPlease put your final answer in \\boxed{}."
}
] | 32 | {
"ground_truth": "32",
"data_type": "klear_math",
"num_tokens": 61
} |
[
{
"role": "user",
"content": "B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let\n$$\n\\frac{d^{1992}}{d x^{1992}}\\left(\\frac{p(x)}{x^{3}-x}\\right)=\\frac{f(x)}{g(x)},\n$$\nwhere $f(x), g(x)$ are polynomials. Find the smallest possible degre... | 3984 | {
"ground_truth": "3984",
"data_type": "klear_math",
"num_tokens": 136
} |
[
{
"role": "user",
"content": "24. When $0<x<\\frac{\\pi}{2}$, the function $y=\\tan 3 x \\cdot \\cot ^{3} x$ cannot take values within the open interval $(a, b)$. Find the value of $a+b$.\n\nPlease put your final answer in \\boxed{}."
}
] | 34 | {
"ground_truth": "34",
"data_type": "klear_math",
"num_tokens": 76
} |
[
{
"role": "user",
"content": "For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\\sqrt{p}|<\\frac{1}{2}$. For example, $b(6)=2$ and $b(23)=5$. Find $S=\\sum_{p=1}^{2007} b(p)$. Please put your final answer in \\boxed{}."
}
] | 59955 | {
"ground_truth": "59955",
"data_type": "klear_math",
"num_tokens": 97
} |
[
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB = 8$ and $BC = 13$. Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \\parallel BC$. Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \\parallel AB$. Find the area of quadrilateral $P_1Q_1P_2Q_2$. Please put your final answer in \\boxed{... | 52 | {
"ground_truth": "52",
"data_type": "klear_math",
"num_tokens": 125
} |
[
{
"role": "user",
"content": "Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$. Please put your final answer in \\boxed{}."
}
] | 86 | {
"ground_truth": "86",
"data_type": "klear_math",
"num_tokens": 43
} |
[
{
"role": "user",
"content": "Let \\(\\mathcal{E}\\) be an ellipse with foci \\(A\\) and \\(B\\). Suppose there exists a parabola \\(\\mathcal{P}\\) such that:\n- \\(\\mathcal{P}\\) passes through \\(A\\) and \\(B\\),\n- the focus \\(F\\) of \\(\\mathcal{P}\\) lies on \\(\\mathcal{E}\\),\n- the orthocenter ... | 2402 | {
"ground_truth": "2402",
"data_type": "klear_math",
"num_tokens": 175
} |
[
{
"role": "user",
"content": "There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of t... | 1012 | {
"ground_truth": "1012",
"data_type": "klear_math",
"num_tokens": 171
} |
[
{
"role": "user",
"content": "A sequence $(z_n)$ of complex numbers satisfies the following properties:\n\n$z_1$ and $z_2$ are not real.\n$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\\geq 1$.\n$\\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\\geq 1$.\n$\\left|\\dfrac{z_3}{z_4}\\right|=\\left|\\dfrac{z_4}... | 65536 | {
"ground_truth": "65536",
"data_type": "klear_math",
"num_tokens": 159
} |
[
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,3009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$.\n\nPlease p... | 1203 | {
"ground_truth": "1203",
"data_type": "klear_math",
"num_tokens": 106
} |
[
{
"role": "user",
"content": "The nineteenth question: Find the smallest real number $\\lambda$ such that $\\sum_{i=1}^{100}\\left(a_{i}-a_{i+1}\\right)^{2} \\leq \\lambda\\left(100-\\left(\\sum_{i=1}^{100} a_{i}\\right)\\right)$ holds for any real numbers $a_{1} 、 a_{2} 、 \\ldots 、 a_{100}$ satisfying $\\s... | 8 | {
"ground_truth": "8",
"data_type": "klear_math",
"num_tokens": 158
} |
[
{
"role": "user",
"content": "Determine the smallest real number \\( \\ell \\) such that every hexagon \\( H \\) in the family \\( \\mathcal{F} \\) can be covered by a strip of width \\( \\ell \\). The family \\( \\mathcal{F} \\) consists of hexagons with parallel opposite sides, and any three vertices of \... | 2 | {
"ground_truth": "2",
"data_type": "klear_math",
"num_tokens": 106
} |
[
{
"role": "user",
"content": "Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.\n\nPlease... | 1594 | {
"ground_truth": "1594",
"data_type": "klear_math",
"num_tokens": 109
} |
[
{
"role": "user",
"content": "Find the constant $k$ such that the sum of all $x \\ge 0$ satisfying $\\sqrt{x}(x+12)=17x-k$ is $256.$\n\nPlease put your final answer in \\boxed{}."
}
] | 216 | {
"ground_truth": "216",
"data_type": "klear_math",
"num_tokens": 60
} |
[
{
"role": "user",
"content": "There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draw... | 45853 | {
"ground_truth": "45853",
"data_type": "klear_math",
"num_tokens": 138
} |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow +\\infty} (xD)^m\\sum_{k=1}^\\infty (-1)^k \\frac{x^k}{k^{k}} \\] where \\(m\\) is a positive integer, \\(D=\\frac{d}{dx}\\), and \\(\\begin{Bmatrix} m\\\\r \\end{Bmatrix}\\) are Stirling numbers of the second kind.\n\nPlease put you... | 0 | {
"ground_truth": "0",
"data_type": "klear_math",
"num_tokens": 113
} |
[
{
"role": "user",
"content": "Given \\( n \\) line segments in space, no three of which are parallel to the same plane, and for which the midpoint connection between any two segments is the common perpendicular of those segments, find the maximum possible value of \\( n \\).\n\nPlease put your final answer ... | 2 | {
"ground_truth": "2",
"data_type": "klear_math",
"num_tokens": 69
} |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\left( x^2 \\cos^2 x - \\sin^2 x \\right)\\left( x^3 - \\sin^3 x \\right)}{\\left( e^x + e^{-x} - x^2 - 2 \\right)^2 \\sin x} \\]\n\nPlease put your final answer in \\boxed{}."
}
] | -48 | {
"ground_truth": "-48",
"data_type": "klear_math",
"num_tokens": 99
} |
[
{
"role": "user",
"content": "A parametric graph is given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{2}, \\\\\ny &= \\sin t.\n\\end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$?\n\nPlease put your final answer in \\boxed{}."
}
] | 12 | {
"ground_truth": "12",
"data_type": "klear_math",
"num_tokens": 87
} |
[
{
"role": "user",
"content": "6. 27 The function $f(k)$ is defined on $N$, taking values in $N$, and is a strictly increasing function (if for any $x_{1}, x_{2} \\in A$, when $x_{1}<x_{2}$, we have $f\\left(x_{1}\\right)<f\\left(x_{2}\\right)$, then $f(x)$ is called a strictly increasing function on $A$), a... | 197 | {
"ground_truth": "197",
"data_type": "klear_math",
"num_tokens": 149
} |
[
{
"role": "user",
"content": "8. A scientist invented a time machine, which looks like a circular subway track. Now (2010) is the first platform, the 2nd, 3rd, ..., 2009th platforms are 2011, 2012, ..., 4018, and the 2010th platform returns to the present (the departure platform). Later, the machine had a p... | 812 | {
"ground_truth": "812",
"data_type": "klear_math",
"num_tokens": 287
} |
[
{
"role": "user",
"content": "99 identical balls lie on a table. 50 balls are made of copper, and 49 balls are made of zinc. The assistant numbered the balls. Once a spectrometer test is applied to 2 balls, it allows determining whether they are made of the same metal or not. However, the results of the tes... | 98 | {
"ground_truth": "98",
"data_type": "klear_math",
"num_tokens": 118
} |
[
{
"role": "user",
"content": "In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceedin... | 3 | {
"ground_truth": "3",
"data_type": "klear_math",
"num_tokens": 109
} |
[
{
"role": "user",
"content": "Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$? Please put your final answer in \\boxed{}."
}
] | 444 | {
"ground_truth": "444",
"data_type": "klear_math",
"num_tokens": 64
} |
[
{
"role": "user",
"content": "If the integers $1,2,\\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, find the largest possible value of $n$.\n\nPlease put your final answer in \\boxed{}."
}
] | 8 | {
"ground_truth": "8",
"data_type": "klear_math",
"num_tokens": 65
} |
[
{
"role": "user",
"content": "A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now t... | 0.5 | {
"ground_truth": "0.5",
"data_type": "klear_math",
"num_tokens": 143
} |
[
{
"role": "user",
"content": "Arrange the numbers 1 through 9 in a 3x3 grid such that consecutive numbers are adjacent (share an edge), and the sum of the numbers in the four corners is 18. What is the number in the center of the grid?\n\nPlease put your final answer in \\boxed{}."
}
] | 7 | {
"ground_truth": "7",
"data_type": "klear_math",
"num_tokens": 72
} |
[
{
"role": "user",
"content": "Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\\frac{1}{3}$ of the way to his destination. Find the number of minutes it... | 245 | {
"ground_truth": "245",
"data_type": "klear_math",
"num_tokens": 107
} |
[
{
"role": "user",
"content": "给定正整数 $n$, 求最大的正整数 $k$, 使得如下命题成立: 对每个 $1 \\leq i \\leq 2 n$,设 $A_{i}$ 是若干个相邻的整数构成的集合（即每个 $A_{i}$ 都形如 $\\{a+1, a+2, \\cdots, a+r\\}$ ，其中 $a$ 是整数, $r$ 是正整数），如果对任何 $1 \\leq i \\leq n, n+1 \\leq j \\leq 2 n$ 都有 $A_{i} \\cap A_{j} \\neq \\emptyset$ ，则存在整数 $x$ ，使得集合 $\\left\\{1 \\leq... | 6 | {
"ground_truth": "6",
"data_type": "klear_math",
"num_tokens": 223
} |
[
{
"role": "user",
"content": "Suppose n is a product of three primes $p_1, p_2, p_3$ where $p_1 < p_2 < p_3$ and $p_1$ is a two-digit integer.\nIf $n - 1$ is a perfect square, compute the smallest possible value of $n$.\n\nPlease put your final answer in \\boxed{}."
}
] | 30277 | {
"ground_truth": "30277",
"data_type": "klear_math",
"num_tokens": 90
} |
[
{
"role": "user",
"content": "Cyclic quadrilateral $ABCD$ satisfies $\\angle ABD = 70^\\circ$ , $\\angle ADB=50^\\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\\angle APQ-\\angle AQP$ .\n\nPlease put your final answer ... | 20 | {
"ground_truth": "20",
"data_type": "klear_math",
"num_tokens": 115
} |
[
{
"role": "user",
"content": "Determine the value of \\( f(k) = \\lim_{n \\rightarrow \\infty} \\frac{|S_n(k)|}{\\pi(n)} \\), where \\( S_n(k) = \\{1 \\leq m \\leq n : m \\text{ has } k \\text{ ones in its binary representation and } m \\text{ is prime}\\} \\) and \\( \\pi(x) \\) is the prime counting funct... | 0 | {
"ground_truth": "0",
"data_type": "klear_math",
"num_tokens": 114
} |
[
{
"role": "user",
"content": "Suppose we have 2013 piles of coins, with the $i^{th}$ pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile... | 11 | {
"ground_truth": "11",
"data_type": "klear_math",
"num_tokens": 117
} |
[
{
"role": "user",
"content": "Suppose $A$ is a non-unital $C^{\\ast}$-algebra. Determine whether there exists an element $a \\in A$ such that $ab = \\lambda b$ for each $b \\in A$, where $\\lambda$ is some complex scalar.\n\nPlease put your final answer in \\boxed{}."
}
] | 0 | {
"ground_truth": "0",
"data_type": "klear_math",
"num_tokens": 78
} |
[
{
"role": "user",
"content": "Let an \"operation\" be randomly changing a given positive integer $n$ to a smaller nonnegative integer (each number has the same probability). What is the probability that after performing the operation on 2019 several times to obtain 0, the numbers 10, 100, and 1000 all appea... | \frac{1}{2019000000} | {
"ground_truth": "\\frac{1}{2019000000}",
"data_type": "klear_math",
"num_tokens": 94
} |
[
{
"role": "user",
"content": "Example 2. Find the smallest positive integer $n$, such that in any 9-vertex and $n$-edge graph with edges colored in two colors, there must exist a monochromatic triangle.\n\n untranslated portion: \n(There is no additional text to translate, as the provided text has been full... | 33 | {
"ground_truth": "33",
"data_type": "klear_math",
"num_tokens": 86
} |
[
{
"role": "user",
"content": "Let \\( M \\) be a finite set of numbers. It is known that among any three elements of this set, there exist two whose sum also belongs to \\( M \\).\n\nWhat is the maximum possible number of elements in \\( M \\)? Please put your final answer in \\boxed{}."
}
] | 7 | {
"ground_truth": "7",
"data_type": "klear_math",
"num_tokens": 69
} |
[
{
"role": "user",
"content": "Given the cubic equation \\( x^3 - 3x + 1 = 0 \\), with roots \\( a < b < c \\), find the value of \\( \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a} \\) without solving the equation.\n\nPlease put your final answer in \\boxed{}."
}
] | -6 | {
"ground_truth": "-6",
"data_type": "klear_math",
"num_tokens": 85
} |
[
{
"role": "user",
"content": "II. (16 points) Let the side lengths opposite to two interior angles of $\\triangle A B C$ be $a, b, c$ respectively, and $a+b+c=16$. Find\n$$\nb^{2} \\cos ^{2} \\frac{C}{2}+c^{2} \\cos ^{2} \\frac{B}{2}+2 b c \\cos \\frac{B}{2} \\cdot \\cos \\frac{C}{2} \\cdot \\sin \\frac{A}{... | 64 | {
"ground_truth": "64",
"data_type": "klear_math",
"num_tokens": 142
} |
[
{
"role": "user",
"content": "How many distinct equilateral triangles can be constructed by connecting three different vertices of a regular dodecahedron?\n\nPlease put your final answer in \\boxed{}."
}
] | 60 | {
"ground_truth": "60",
"data_type": "klear_math",
"num_tokens": 43
} |
[
{
"role": "user",
"content": "If Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the form \\(\\frac{m}{n}\\). Please calculate the value of m + n. Please put your final answer in \\boxed{}."
}
] | 1717 | {
"ground_truth": "1717",
"data_type": "klear_math",
"num_tokens": 68
} |
[
{
"role": "user",
"content": "The sequence $\\{a_n\\}_{n\\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$, with initial values $a_1 = 1$ and $a_2 = 8$. Another sequence, $\\{b_n\\}$, is defined by the recurrence relation $b_{n+2} = 3b_{n+1} - b_n$ for posit... | 89 | {
"ground_truth": "89",
"data_type": "klear_math",
"num_tokens": 166
} |
[
{
"role": "user",
"content": "A palindrome is a word that reads the same backward as forward. Examples include OMO, lepel, and parterretrap. Determine how many palindromes can be constructed using the letters $a, b, c, d,$ and $e$ under the following conditions:\n\n- Each letter may appear no more than twic... | 640 | {
"ground_truth": "640",
"data_type": "klear_math",
"num_tokens": 120
} |
[
{
"role": "user",
"content": "The vertices of a cube are labeled with the integers 1 through 8, with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.\n\nPlease put your final answer in \\boxed{}... | 11 | {
"ground_truth": "11",
"data_type": "klear_math",
"num_tokens": 76
} |
[
{
"role": "user",
"content": "Find the smallest $n \\in \\mathbb{N}$ such that if any 5 vertices of a regular $n$ -gon are colored red, there exists a line of symmetry $l$ of the $n$ -gon such that every red point is reflected across $l$ to a non-red point.\n\nPlease put your final answer in \\boxed... | 14 | {
"ground_truth": "14",
"data_type": "klear_math",
"num_tokens": 93
} |
[
{
"role": "user",
"content": "80. Three students' scores in a math exam are all different, and none of them scored 0 or 100. Each student knows their own score and ranking, but not the scores or rankings of the other two. They then provide information to each other:\n甲 says: “My score is a multiple of 10.”\... | 84 | {
"ground_truth": "84",
"data_type": "klear_math",
"num_tokens": 150
} |
[
{
"role": "user",
"content": "If the coefficient of the third term in the binomial expansion of $(1 - 3x)^{1/4}$ is $-\\frac{a}{b}$, where $a$ and $b$ are relatively prime integers, find $a + b$. Please put your final answer in \\boxed{}."
}
] | 59 | {
"ground_truth": "59",
"data_type": "klear_math",
"num_tokens": 73
} |
[
{
"role": "user",
"content": "在凸四边形 $A B C D$ 的外部分别作正 $\\triangle A B Q 、 \\triangle B C R 、 \\triangle C D S 、 \\triangle D A P$, 记四边形 $A B C D$ 的对角线的和为 $x$, 四边形 $P Q R S$ 的对边中点连线的和为 $y$. 求 $\\frac{y}{x}$ 的最大值。原始答案为 $\\frac{a+\\sqrt{b}}{c}$ 的格式，请给出 a + b + c的值。\n\nPlease put your final answer in \\boxed{}.... | 6 | {
"ground_truth": "6",
"data_type": "klear_math",
"num_tokens": 137
} |
[
{
"role": "user",
"content": "When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\\leq 2007$ and t... | 14 | {
"ground_truth": "14",
"data_type": "klear_math",
"num_tokens": 114
} |
[
{
"role": "user",
"content": "Determine the smallest integer $k > 1$ such that in a $k$-player tournament, every player plays every other player exactly once, and for any two players, there exists a third player who defeated both. Provide a proof for your answer.\n\nPlease put your final answer in \\boxed{}... | 7 | {
"ground_truth": "7",
"data_type": "klear_math",
"num_tokens": 73
} |
[
{
"role": "user",
"content": "In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle... | 44 | {
"ground_truth": "44",
"data_type": "klear_math",
"num_tokens": 84
} |
[
{
"role": "user",
"content": "In their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. Find the sum of the digits of the base 10 representation of $9ab$.\n\nPlease put your final answer in \\boxed{}."
}
] | 17865 | {
"ground_truth": "17865",
"data_type": "klear_math",
"num_tokens": 85
} |
[
{
"role": "user",
"content": "Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$ $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$ *Proposed by Kyle Lee*\n\nPlease put yo... | 129 | {
"ground_truth": "129",
"data_type": "klear_math",
"num_tokens": 132
} |
[
{
"role": "user",
"content": "8. 1b.(TUR 5) Find the smallest positive integer $n$ such that (i) $n$ has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of $n$.\n\nPlease put your final answer in \\boxed{}."
}
] | 110880 | {
"ground_truth": "110880",
"data_type": "klear_math",
"num_tokens": 75
} |
[
{
"role": "user",
"content": "In a mathematics competition, 10 students each solve exactly 3 problems. Any pair of students solves at least 1 common problem. Determine the minimum number of students who solve a common problem that is solved by the most students.\n\nPlease put your final answer in \\boxed{}.... | 5 | {
"ground_truth": "5",
"data_type": "klear_math",
"num_tokens": 67
} |
[
{
"role": "user",
"content": "6. Given a positive integer $k$ that satisfies for any positive integer $n$, the smallest prime factor of $n^{2}+n-k$ is no less than 11. Then, $k_{\\text {min }}$ $=$ . $\\qquad$\n\nPlease put your final answer in \\boxed{}."
}
] | 43 | {
"ground_truth": "43",
"data_type": "klear_math",
"num_tokens": 78
} |
[
{
"role": "user",
"content": "Let $n$ be an integer greater than 1, and let $X$ be a set of $n$ elements. If the union $A_{1} \\cup \\cdots \\cup A_{k}$ is a proper subset of $X$, and no element of $X$ is in exactly one of $A_{i}$, then the set of non-empty subsets $A_{1}, \\ldots, A_{k}$ of $X$ is compact.... | 198 | {
"ground_truth": "198",
"data_type": "klear_math",
"num_tokens": 156
} |
[
{
"role": "user",
"content": "A triple of positive integers $(a, b, c)$ is tasty if $\\operatorname{lcm}(a, b, c) \\mid a+b+c-1$ and $a<b<c$. Find the sum of $a+b+c$ across all tasty triples.\n\nPlease put your final answer in \\boxed{}."
}
] | 44 | {
"ground_truth": "44",
"data_type": "klear_math",
"num_tokens": 75
} |
[
{
"role": "user",
"content": "Dragon selects three positive real numbers with sum $100$, uniformly at random. He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription. If the probability the three new numbers still sum to $100$ is $\\tfrac{m}{n}$, wher... | 704 | {
"ground_truth": "704",
"data_type": "klear_math",
"num_tokens": 120
} |
[
{
"role": "user",
"content": "7.7. Consider the triangle $A B C$ with $A B=A C$ and $m(\\angle B)>30^{\\circ}$. Inside the triangle, we have a point $M$ such that $m(\\angle M B C)=30^{\\circ}$ and $m(\\angle M A B)=\\frac{3}{4} m(\\angle B A C)$. Determine $m(\\angle A M C)$.\n\nPlease put your final answe... | 150 | {
"ground_truth": "150",
"data_type": "klear_math",
"num_tokens": 114
} |
[
{
"role": "user",
"content": "There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select three members of that club to give a copy of $\\emph{Introduction to} \\allowbreak\\ \\emph{Counting and... | 71 | {
"ground_truth": "71",
"data_type": "klear_math",
"num_tokens": 152
} |
[
{
"role": "user",
"content": "Determine the number of ordered triples of complex numbers \\((x, y, z)\\) such that \\(x^2 - y^3\\), \\(y^2 - z^3\\), and \\(z^2 - x^3\\) are rational numbers, and \\(x^3(x^2+1) + y^3(y^2+1) + z^3(z^2+1) = 2[(x^2)y^3 + (y^2)z^3 + (z^2)x^3]\\).\n\nPlease put your final answer i... | 9 | {
"ground_truth": "9",
"data_type": "klear_math",
"num_tokens": 135
} |
[
{
"role": "user",
"content": "Determine the maximum value of $m^{2} + n^{2}$, where $m, n \\in \\{1, 2, \\cdots, 1981\\}$, and $(n^{2} - mn - m^{2})^{2} = 1$.\n\nPlease put your final answer in \\boxed{}."
}
] | 3524578 | {
"ground_truth": "3524578",
"data_type": "klear_math",
"num_tokens": 83
} |
[
{
"role": "user",
"content": "In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of ... | 176 | {
"ground_truth": "176",
"data_type": "klear_math",
"num_tokens": 155
} |
[
{
"role": "user",
"content": "Let \\(\\alpha\\) and \\(\\beta\\) be the roots of the equations \\(\\log _{2} x + x + 2 = 0\\) and \\(2^{x} + x + 2 = 0\\) respectively. Find the value of \\(\\alpha + \\beta\\).\n\nPlease put your final answer in \\boxed{}."
}
] | -2 | {
"ground_truth": "-2",
"data_type": "klear_math",
"num_tokens": 82
} |
[
{
"role": "user",
"content": "In an isosceles triangle $A B C \\quad \\angle B=\\operatorname{arctg} 8 / 15$. A circle of radius 1, inscribed in angle $C$, touches side $C B$ at point $M$ and cuts off a segment $K E$ from the base. It is known that $M B=15 / 8$. Find the area of triangle $K M B$, given that... | \frac{375}{272} | {
"ground_truth": "\\frac{375}{272}",
"data_type": "klear_math",
"num_tokens": 138
} |
[
{
"role": "user",
"content": "Two trains are 80 kilometers apart and are traveling toward each other on the same track, each at a speed of 25 kilometers per hour. A bee starts at the first train and flies back and forth between the two trains until they are 100 meters apart. The bee completes 36 rounds, lan... | 270 | {
"ground_truth": "270",
"data_type": "klear_math",
"num_tokens": 120
} |
[
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radiu... | 134 | {
"ground_truth": "134",
"data_type": "klear_math",
"num_tokens": 167
} |
[
{
"role": "user",
"content": "$\\stackrel{\\text { Senderov B.A. }}{ }$.\n\nFind the smallest natural number that cannot be represented in the form $\\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are natural numbers.\n\nPlease put your final answer in \\boxed{}."
}
] | 11 | {
"ground_truth": "11",
"data_type": "klear_math",
"num_tokens": 83
} |
[
{
"role": "user",
"content": "In a tournament, each participant plays a match against each other participant. The winner of a match earns 1 point, the loser 0, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (if m... | 25 | {
"ground_truth": "25",
"data_type": "klear_math",
"num_tokens": 122
} |
[
{
"role": "user",
"content": "How many distinct values of $x$ satisfy \n$\\lfloor{x}\\rfloor^2-3x+2=0$, where $\\lfloor{x}\\rfloor$ denotes the largest integer less than or equal to $x$? Please put your final answer in \\boxed{}."
}
] | 4 | {
"ground_truth": "4",
"data_type": "klear_math",
"num_tokens": 68
} |
[
{
"role": "user",
"content": "Berroov S.L.\n\nOne hundred integers are written in a circle. Each number is greater than the sum of the two numbers following it in a clockwise direction.\n\nWhat is the maximum number of positive numbers that can be among those written?\n\n#\n\nPlease put your final answer in... | 49 | {
"ground_truth": "49",
"data_type": "klear_math",
"num_tokens": 72
} |
[
{
"role": "user",
"content": "Determine the number of 8-tuples in the set $[4]^8 = \\{1, 2, 3, 4\\}^8$ such that each number from $[4]$ appears exactly twice and no number occupies its corresponding index (e.g., $i$ does not appear in the $i$th position for all $i \\in [4]$). Is this number equal to $8!/16$... | 864 | {
"ground_truth": "864",
"data_type": "klear_math",
"num_tokens": 131
} |
[
{
"role": "user",
"content": "M . B.\n\nOn some cells of a $10 \\times 10$ board, there is a flea. Once a minute, the fleas jump simultaneously, and each one jumps to an adjacent cell (along a side). A flea jumps strictly in one of the four directions parallel to the sides of the board, maintains its direct... | 40 | {
"ground_truth": "40",
"data_type": "klear_math",
"num_tokens": 146
} |
[
{
"role": "user",
"content": "32. [17] Suppose point $P$ is inside triangle $A B C$. Let $A P, B P$, and $C P$ intersect sides $B C, C A$, and $A B$ at points $D, E$, and $F$, respectively. Suppose $\\angle A P B=\\angle B P C=\\angle C P A, P D=\\frac{1}{4}, P E=\\frac{1}{5}$, and $P F=\\frac{1}{7}$. Compu... | \frac{19}{12} | {
"ground_truth": "\\frac{19}{12}",
"data_type": "klear_math",
"num_tokens": 137
} |
[
{
"role": "user",
"content": "For the NEMO, Kevin needs to compute the product\n\t\t\\[\n\t\t9 \\times 99 \\times 999 \\times \\cdots \\times 999999999.\n\t\t\\]\n\t\tKevin takes exactly $ab$ seconds to multiply an $a$ -digit integer by a $b$ -digit integer. Compute the minimum number of seconds necessa... | 870 | {
"ground_truth": "870",
"data_type": "klear_math",
"num_tokens": 135
} |
[
{
"role": "user",
"content": "How many subsets of the set $\\{1, 2, 3, \\ldots, 12\\}$ contain exactly one or two prime numbers? Please put your final answer in \\boxed{}."
}
] | 1920 | {
"ground_truth": "1920",
"data_type": "klear_math",
"num_tokens": 52
} |
[
{
"role": "user",
"content": "Suppose $ 2015= a_1 <a_2 < a_3<\\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \\in \\mathbb{N} $ and $1\\le m,n \\le k $ , $$ a_m+a_n\\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.\n\nPlease put your final ... | 2016 | {
"ground_truth": "2016",
"data_type": "klear_math",
"num_tokens": 124
} |
[
{
"role": "user",
"content": "Find the number of positive integers less than 100 that are divisors of 300. Please put your final answer in \\boxed{}."
}
] | 15 | {
"ground_truth": "15",
"data_type": "klear_math",
"num_tokens": 41
} |
[
{
"role": "user",
"content": "设 a_1,a_2,a_3,a_4 是正整数，满足在它们的任意圆排列中，均存在相邻两项不互质. 求三元组 (i,j,k) 个数的最大值，使得 \\{i,j,k\\}\\subseteq\\{1,2,3,4\\}，且 (a_i,a_j)^2\\mid a_k.\n\nPlease put your final answer in \\boxed{}."
}
] | 16 | {
"ground_truth": "16",
"data_type": "klear_math",
"num_tokens": 102
} |
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