prompt listlengths 1 1 | reward_model dict | data_source stringclasses 2
values | ability stringclasses 1
value |
|---|---|---|---|
[
{
"content": "N numbers are marked in the set $\\{1,2,...,2000\\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$.\n\nI.Gorodnin",
"role": "user"
}
] | {
"ground_truth": "666",
"mean_reward": 0.359375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The following is a math problem and a solution (split into paragraphs, enclosed with tags and indexed from 0):\n\n[Math Problem]\n\nLet $m \\neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \n\\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(... | {
"ground_truth": "2",
"mean_reward": 0.078125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "For how many positive integers $n \\le 500$ is $n!$ divisible by $2^{n-2}$?\n\n[i]Proposed by Eugene Chen[/i]",
"role": "user"
}
] | {
"ground_truth": "44",
"mean_reward": 0.359375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "In the city of liars and knights, there are 366 inhabitants, each born on a different day of a leap year. All inhabitants of the city answered two questions. To the question \"Were you born in February?\" 100 people answered affirmatively, and to the question \"Were you born on the 30th day?\" 60 ... | {
"ground_truth": "29",
"mean_reward": 0.125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds of the Chinese people... | {
"ground_truth": "72",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "6. let $A, B, C$ and $D$ be four points lying in this order on a circle. Assume that there is a point $K$ on the line $A B$ such that $B D$ bisects the line $K C$ and $A C$ bisects the line $K D$. Determine the smallest possible value that $\\left|\\frac{A B}{C D}\\right|$ can take.",
"role": ... | {
"ground_truth": "2",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "5. If in an acute scalene triangle three medians, three angle bisectors, and three altitudes are drawn, they will divide it into 34 parts.",
"role": "user"
}
] | {
"ground_truth": "34",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "What is the value of $\\sqrt{2 \\cdot 4! \\cdot 4!}$ expressed as a positive integer?",
"role": "user"
}
] | {
"ground_truth": "24\\sqrt{2}",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Determine all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ with the property\n\n$$\nf(f(x)+y)=2 x+f(f(y)-x)\n$$\n\nfor all $x, y \\in \\mathbb{R}$.",
"role": "user"
}
] | {
"ground_truth": "f(x)=x+c",
"mean_reward": 0.0625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "62. Given that the volume of a rectangular prism is 20 cubic centimeters, and the length, width, and height are all whole centimeters, there are $\\qquad$ such rectangular prisms.",
"role": "user"
}
] | {
"ground_truth": "4",
"mean_reward": 0.1875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "3. It is known that the polynomial $f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4}$ has 4 distinct real roots $\\left\\{x_{1}, x_{2}, x_{3}, x_{4}\\right\\}$. The polynomial of the form $g(x)=b_{0}+b_{1} x+b_{2} x^{2}+b_{3} x^{3}+x^{4}$ has roots $\\left\\{x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}\\right\\}$... | {
"ground_truth": "-1216",
"mean_reward": 0.359375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Mr. Thompson recorded the following information from a science test taken by $100$ students in his classes:\n\\begin{tabular}{|c|c|}\n\\multicolumn{2}{c}{}\\\\\\hline\n\\textbf{$\\%$ Score}&\\textbf{Number of Students}\\\\\\hline\n100&4\\\\\\hline\n90&10\\\\\\hline\n80&30\\\\\\hline\n70&19\\\\\\hl... | {
"ground_truth": "69.7",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30... | {
"ground_truth": "12",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations?",
"role": "user"
}
] | {
"ground_truth": "55",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives... | {
"ground_truth": " n = 2^k ",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "7. Given $|\\boldsymbol{a}|=1,|\\boldsymbol{b}|=|c|=2, \\boldsymbol{b} \\cdot \\boldsymbol{c}=0$, $\\lambda \\in(0,1)$. Then\n$$\n|a-b+\\lambda(b-c)|+\\left|\\frac{1}{2} c+(1-\\lambda)(b-c)\\right|\n$$\n\nthe minimum value is $\\qquad$",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}-1",
"mean_reward": 0.359375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "10.8. On a plane, $N$ points are marked. Any three of them form a triangle, the angles of which in degrees are expressed by natural numbers. For what largest $N$ is this possible?",
"role": "user"
}
] | {
"ground_truth": "180",
"mean_reward": 0.328125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$",
"role": "user"
}
] | {
"ground_truth": "40",
"mean_reward": 0.171875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $S$ is inside $\\triangle PQR$, $\\angle PQR = 50^\\circ$, and $\\angle PRS = 110^\\circ$. What is the degree measure of $\\angle\nPQS$?",
"role": "user"
}
] | {
"ground_truth": "30^\\circ",
"mean_reward": 0.234375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$ \\exp \\left( \\sum_{n=1}^{\\infty} \\frac{ B(n)}{n(n+1)} \\right) $$ is a rational number.",
"role": "user"
}
] | {
"ground_truth": " 4 ",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all different from each other. What is the minimum number of distinct numbers in \\( a_{1}, a_{2}, \\cd... | {
"ground_truth": "46",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?",
"role": "user"
}
] | {
"ground_truth": "90",
"mean_reward": 0.109375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "\nTwo girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is... | {
"ground_truth": "5\\%",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "68 Mark 10 points on a circle, the number of convex polygons that can be formed with these points is $\\qquad$.",
"role": "user"
}
] | {
"ground_truth": "968",
"mean_reward": 0.046875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let $a_1$ , $a_2$ , $\\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\\gcd(a_{k-1},a_k)$ for all $k\\geq 2$ . Compute the smallest possible value of $a_1+a_2+\\cdots+a_{10}$ .",
"role": "user"
}
] | {
"ground_truth": "440",
"mean_reward": 0.046875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "In a corridor that is 100 meters long, there are 20 rugs with a total length of 1 kilometer. Each rug is as wide as the corridor. What is the maximum possible total length of the sections of the corridor that are not covered by the rugs?",
"role": "user"
}
] | {
"ground_truth": "50",
"mean_reward": 0.34375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Example 3 For a set of numbers $M$, define the sum of $M$ as the sum of all numbers in $M$, denoted as $S(M)$. Let $M$ be a set composed of several positive integers not greater than 15, and any two disjoint subsets of $M$ have different sums. Find the maximum value of $S(M)$.",
"role": "user"... | {
"ground_truth": "61",
"mean_reward": 0.109375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "How many times longer is the staircase to the fourth floor of a building compared to the staircase to the second floor of the same building?",
"role": "user"
}
] | {
"ground_truth": "3",
"mean_reward": 0.15625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Positive $x,y,z$ satisfy a system: $\\begin{cases} x^2 + xy + y^2/3= 25 \\\\\n y^2/ 3 + z^2 = 9 \\\\\nz^2 + zx + x^2 = 16 \\end{cases}$\n\nFind the value of expression $xy + 2yz + 3zx$.",
"role": "user"
}
] | {
"ground_truth": "24\\sqrt{3}",
"mean_reward": 0.15625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "\\section*{Problem 1 - 101211}\n\nIn a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending.\n\nOf exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boys and exactly 3 gi... | {
"ground_truth": "4",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Given a $54^{\\circ}$ angle, divide it into three equal parts using a compass and straightedge.",
"role": "user"
}
] | {
"ground_truth": "18^\\circ",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "5,6,7 |\n\nGiuseppe has a sheet of plywood measuring $22 \\times 15$. Giuseppe wants to cut as many rectangular blanks of size $3 \\times 5$ as possible from it. How can he do this?",
"role": "user"
}
] | {
"ground_truth": "22",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Example 12 Convert $(111001101010)_2$ to an octal number.\n\nTranslate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.",
"role": "user"
}
] | {
"ground_truth": "(7152)_8",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\\sqrt{d} - b$ cm, where $d$ is a prime number. Find the value of $b + d$.",
"role": "user"
}
] | {
"ground_truth": "12.5",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\\leq x \\leq 15$. The probability that $\\lfloor \\sqrt{P(x)} \\rfloor = \\sqrt{P(\\lfloor x \\rfloor )}$ is equal to $\\dfrac{\\sqrt{a}+\\sqrt{b}+\\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and non... | {
"ground_truth": "850",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Example 3 There is a railway network between six cities, such that there is a direct railway between any two cities. On Sundays, some railways will be closed for maintenance. The railway department stipulates: after closing several sections of the railway, it must still be possible to travel by ra... | {
"ground_truth": "26703",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "1. Given an acute triangle $\\triangle A B C$ with three interior angles satisfying $A>B>C$, let $\\alpha$ denote the maximum of $90^{\\circ}-A, A-B, B-C$. Find the minimum value of $\\alpha$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and o... | {
"ground_truth": "15^{\\circ}",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.\nWhat is the maximal possible number of points with this... | {
"ground_truth": "8",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "In a certain meeting, there are 30 participants. Each person knows at most 5 others among the remaining participants, and in any group of 5 people, there are always two who do not know each other. Find the largest positive integer \\( k \\) such that there exists a subset of \\( k \\) people from ... | {
"ground_truth": "6",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Example 4 Let integer $n \\geqslant 3, a_{1}, a_{2}, \\cdots, a_{n}$ be any $n$ distinct real numbers with a positive sum; $b_{1}, b_{2}, \\cdots, b_{n}$ is a permutation of these $n$ numbers. If for $k=1,2, \\cdots, n$, we have $\\sum_{i=1}^{k} b_{i}>0$, then $b_{1}, b_{2}, \\cdots, b_{n}$ is cal... | {
"ground_truth": "(n-1)!",
"mean_reward": 0.140625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A woman was born in the nineteenth century and was $x$ years old in the year $x^2$. Find the birth year of the woman.",
"role": "user"
}
] | {
"ground_truth": "1892",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "## Task 6A - 211236A\n\n\n\nConsider a stack of items (such as cans) arranged as shown in the figure, where for $k=1,2, \\ldots, m$, the $k$-th row contains exactly $k$ ... | {
"ground_truth": "7140",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.\r\n\r\n(i) One may give cookies only to people adjacent to himself.\r\n(ii) In order to give a cookie to one's ... | {
"ground_truth": "2^n",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Sides $\\overline{AB}$ and $\\overline{BC}$ of regular pentagon $ABCDE$ are extended to meet at point $P$. What is the degree measure of angle $P$?",
"role": "user"
}
] | {
"ground_truth": "36^\\circ",
"mean_reward": 0.046875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "There are 12 items displayed on a shelf, arranged with 4 items on the upper layer and 8 items on the lower layer. Now, if 2 items are to be selected from the lower layer to be moved to the upper layer, and the relative order of the other items remains unchanged, calculate the number of different w... | {
"ground_truth": "840",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$\n\n[asy]\nunitsize(2 cm);\n\npair A, B, C, Ap, Bp... | {
"ground_truth": "140",
"mean_reward": 0.34375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The probability of event \\( A \\) occurring in each of 484 independent trials is 0.75. Find the probability that the relative frequency of event \\( A \\) deviates from its probability by no more than 0.03 in absolute value.",
"role": "user"
}
] | {
"ground_truth": "0.872",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "# Problem №8 (15 points)\n\nA load was decided to be weighed on unequal-arm scales. When the load was placed on one of the pans of these scales, a weight of mass \\( m_{1}=0.5 \\) kg had to be placed on the other side to balance it. In the situation where the load was placed on the other pan of th... | {
"ground_truth": "0.875",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018} \\cdot a_{2020} \\cdot a_{2022}$.",
"role": "user"
}
] | {
"ground_truth": "112",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "2. Vasya and Petya, participating in a school sports and entertainment game, need to cover a distance of 3 km as quickly as possible with only one pair of roller skates between them. They start simultaneously, one just running, the other running on roller skates. At any time, the one running on ro... | {
"ground_truth": "0.5",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "6. Let $\\left(x_{1}, x_{2}, \\cdots, x_{20}\\right)$ be a permutation of $(1,2, \\cdots, 20)$, and satisfy\n$$\n\\sum_{i=1}^{20}\\left(\\left|x_{i}-i\\right|+\\left|x_{i}+i\\right|\\right)=620 \\text {. }\n$$\n\nThen the number of such permutations is. $\\qquad$",
"role": "user"
}
] | {
"ground_truth": "(10!)^{2}",
"mean_reward": 0.265625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Zhendarov R.G.\n\nCells of a $9 \\times 9$ board are painted in a checkerboard pattern with black and white colors (corner cells are white). What is the minimum number of rooks needed to be placed on this board so that all white cells are under attack by these rooks? (A rook attacks all cells in t... | {
"ground_truth": "5",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Find the smallest number $\\mathrm{n}$ such that given any $\\mathrm{n}$ distinct numbers from $\\{1,2,3, \\ldots, 999\\}$, one can choose four different numbers $\\mathrm{a}, \\mathrm{b}, \\mathrm{c}, \\mathrm{d}$ such that $\\mathrm{a}+2 \\mathrm{~b}+3 \\mathrm{c}=\\mathrm{d}$.",
"role": "us... | {
"ground_truth": "835",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Example 8 (2003 Bulgaria National Mathematical Olympiad) Find all real numbers $a$ that satisfy the condition $4[a n]=n+[a[a n]]$, where $n$ is any positive integer.\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result ... | {
"ground_truth": "2+\\sqrt{3}",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "7. In the right figure, each of the three rods of the abacus has 10 beads. By dividing the beads on each rod into upper and lower parts, two three-digit numbers are obtained. The upper part must be a three-digit number with all different digits, and it must be a multiple of the three-digit number ... | {
"ground_truth": "925",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$. The average value of $S$ (if all possible orders of these 20 people are... | {
"ground_truth": "9",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "What is the smallest number $n$ such that one can choose $n$ distinct odd integers (none equal to 1) with \\(\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n}=1\\)?",
"role": "user"
}
] | {
"ground_truth": "9",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "\\section*{Problem 6A - 141236A}\n\nA measurement complex \\(M\\) integrated into an industrial process transmits to a transmission unit \\(A_{1}\\) exactly one of the two signals \\(S_{1}\\) or \\(S_{2}\\), which is then transmitted from \\(A_{1}\\) to a transmission unit \\(A_{2}\\), from \\(A_{... | {
"ground_truth": "0.970596",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let us consider a set $S = \\{ a_1 < a_2 < \\ldots < a_{2004}\\}$ , satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \\quad \\forall i, j$ from $\\{1, 2,\\ldots , 2004\\}$ , where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$ . Find the... | {
"ground_truth": "1003",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "9. Two differentiable real functions $f(x)$ and $g(x)$ satisfy\n$$\n\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}=e^{f(x)-g(x)}\n$$\nfor all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$.",
"role": "user"
}
] | {
"ground_truth": "1-\\ln2",
"mean_reward": 0.21875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "# Task 8.2\n\nFor a natural number $N$, all its divisors were listed, and then the sum of digits for each of these divisors was calculated. It turned out that among these sums, all numbers from 1 to 9 were found. Find the smallest value of $\\mathrm{N}$.\n\n## Number of points 7",
"role": "use... | {
"ground_truth": "288",
"mean_reward": 0.140625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Alex is a young entrepreneur who is excited about the potential of Industry 4.0, particularly in improving efficiency through automation and data analysis. Alex decides to invest in three smart machines for their small manufacturing startup. \n\nThe first machine costs $2,500 and can produce 100 u... | {
"ground_truth": "1235",
"mean_reward": 0.234375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?",
"role": "user"
}
] | {
"ground_truth": "26",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal. Moreover, among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.\n(4th... | {
"ground_truth": "61",
"mean_reward": 0.046875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "6. The Sultan gathered 300 court wise men and proposed a test. There are 25 different colors of hats, known in advance to the wise men. The Sultan informed them that one of these hats would be placed on each of the wise men, and if the number of hats of each color is written down, all the numbers ... | {
"ground_truth": "150",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A six-digit square number, when divided into two-digit segments, reveals that the last two-digit number is the same as the middle two-digit number, while the first two-digit number completes the middle two-digit number to 100. Which is this square number?",
"role": "user"
}
] | {
"ground_truth": "316969",
"mean_reward": 0.171875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 100$ such that $i^x+i^y$ is a real number.",
"role": "user"
}
] | {
"ground_truth": "1850",
"mean_reward": 0.265625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Solve the system of linear equations \\(\\left\\{\\begin{array}{l}{5x-2y=4}\\\\{2x+3y=9}\\end{array}\\right.\\) using the addition and subtraction elimination method to eliminate x. Find the values of a and b such that a * equation 1 + b * equation 2 eliminates x. Express your answer as a pair of ... | {
"ground_truth": "(2, -5)",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "If $\\angle A=20^\\circ$ and $\\angle AFG=\\angle AGF,$ then how many degrees is $\\angle B+\\angle D?$ [asy]\n/* AMC8 2000 #24 Problem */\npair A=(0,80), B=(46,108), C=(100,80), D=(54,18), E=(19,0);\ndraw(A--C--E--B--D--cycle);\nlabel(\"$A$\", A, W);\nlabel(\"$B$ \", B, N);\nlabel(\"$C$\", shift(... | {
"ground_truth": "80^\\circ",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.",
"role": "user"
}
] | {
"ground_truth": "60",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Triangle $A B C$ has incircle $\\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\\omega$ that is closer to... | {
"ground_truth": "14/65",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Problem 6.8. There are exactly 120 ways to color five cells in a $5 \\times 5$ table so that each column and each row contains exactly one colored cell.\n\nThere are exactly 96 ways to color five cells in a $5 \\times 5$ table without a corner cell so that each column and each row contains exactly... | {
"ground_truth": "78",
"mean_reward": 0.390625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "4. A net of a cube is shown with one integer on each face. A larger cube is constructed using 27 copies of this cube. What is the minimum possible sum of all of the integers showing on the six faces of the larger cube?",
"role": "user"
}
] | {
"ground_truth": "90",
"mean_reward": 0.234375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "2. As shown in the figure, Hua Hua washed 40 handkerchiefs and hung them on 3 lines to dry. Each handkerchief must be held at both ends with clips. Hua Hua used $\\qquad$ clips in total.",
"role": "user"
}
] | {
"ground_truth": "43",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "\nYou need to cut a wooden cube with an edge length of 40 cm into 64 smaller cubes with an edge length of 10 cm. This can be easily done with nine cuts if you do not move the cut pieces relative to each other. By repositioning the cut pieces after each cut, how much can you reduce the number of cu... | {
"ground_truth": "6",
"mean_reward": 0.015625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Find all functions \\( f: \\mathbf{N}^{*} \\rightarrow \\mathbf{N}^{*} \\) such that for any \\( x, y \\in \\mathbf{N}^{*} \\), the following holds:\n\\[ f(x + f(y)) = f(x) + y .\\]",
"role": "user"
}
] | {
"ground_truth": "f(x) = x",
"mean_reward": 0.34375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Find all real values of the parameter $a$ for which the system\n\\begin{align*}\n&1+\\left(4x^2-12x+9\\right)^2+2^{y+2}=a\n&\\log_3\\left(x^2-3x+\\frac{117}4\\right)+32=a+\\log_3(2y+3)\n\\end{align*}has a unique real solution. Solve the system for these values of $a$ .",
"role": "user"
}
... | {
"ground_truth": " a = 33 ",
"mean_reward": 0.03125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Please write an irrational number that is greater than -3 and less than -2.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{5}",
"mean_reward": 0.265625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Given that Thomas the painter initially had enough paint for 50 identically sized rooms, and 4 cans of paint were lost, determining how many cans of paint he used for 42 rooms.",
"role": "user"
}
] | {
"ground_truth": "21",
"mean_reward": 0.25
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid.",
"role": "user"
}
] | {
"ground_truth": "18",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "The following is a math problem and a solution (split into paragraphs, enclosed with tags and indexed from 0):\n\n[Math Problem]\n\nDetermine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution.\n\n[Solution]\n\n<paragraph_0>\nTo determine ... | {
"ground_truth": "3",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\\frac{BP}{PC}=\\frac{1}{2}$ . If $\\measuredangle ABC$ $=$ $45^{\\circ}$ and $\\measuredangle APC$ $=$ $60^{\\circ}$ , determine $\\measuredangle ACB$ without trigonometry.",
"role": "user"
}
] | {
"ground_truth": "75^\\circ",
"mean_reward": 0.140625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "An equilateral triangle $ABC$ shares a common side $BC$ with a square $BCDE,$ as pictured. What is the number of degrees in $\\angle DAE$ (not pictured)? [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + dir(300);\npC = pA + dir(240);\npD = pC + dir(270);\npE = pB + dir(270);\ndraw(pA--pB--... | {
"ground_truth": "30^\\circ.",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "How many coins can be placed on a $10 \\times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?",
"role": "user"
}
] | {
"ground_truth": "34",
"mean_reward": 0.0625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "3. Find the area of the region defined by the inequality: $|y-| x-2|+| x \\mid \\leq 4$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.",
"role": "user"
}
] | {
"ground_truth": "32",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "3. For each of the numbers $1,2, \\ldots, 201$, choose one of the signs + or -, so that the expression $A=| \\pm 1 \\pm 2 \\pm \\ldots \\pm 2011|$ has the smallest value.",
"role": "user"
}
] | {
"ground_truth": "0",
"mean_reward": 0.1875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "4. There is an integer greater than 1, which when dividing $357,262,205$ yields the same remainder. Find this integer.",
"role": "user"
}
] | {
"ground_truth": "19",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "11. In a convex hexagon $A B C D E F$ with 6 equal sides, the interior angles $A, B, C$ are $134^{\\circ}, 106^{\\circ}, 134^{\\circ}$, respectively. Then the interior angle $E$ is $\\qquad$\n(Answer in degrees)",
"role": "user"
}
] | {
"ground_truth": "134",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Problem 9.8. On the side $CD$ of trapezoid $ABCD (AD \\| BC)$, a point $M$ is marked. A perpendicular $AH$ is dropped from vertex $A$ to segment $BM$. It turns out that $AD = HD$. Find the length of segment $AD$, given that $BC = 16$, $CM = 8$, and $MD = 9$.\n\n![](https://cdn.mathpix.com/cropped/... | {
"ground_truth": "18",
"mean_reward": 0.296875
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A box contains $900$ cards, labeled from $100$ to $999$ . Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?",
"role": "user"
}
] | {
"ground_truth": "53",
"mean_reward": 0.265625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \\ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\\ldots+a_{n} x_{n}$, where $(a_{1}, \\ldots, a_{n})$ ... | {
"ground_truth": "1",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday? Express your answer as a single year in the format YYYY.",
"role": "user"
}
] | {
"ground_truth": "2017",
"mean_reward": 0.078125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Maria is a young Brazilian lady who recently converted to Catholicism. She is passionate about social justice and fighting racial discrimination. She decided to organize a charity event at her local church to raise funds for a campaign promoting racial equality. \n\nMaria plans to sell handmade br... | {
"ground_truth": "180",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Two circles touch each other externally at point \\( C \\). A line is tangent to the first circle at point \\( A \\) and to the second circle at point \\( B \\). The line \\( AC \\) intersects the second circle at point \\( D \\), distinct from \\( C \\). Find \\( BC \\), given that \\( AC = 9 \\)... | {
"ground_truth": "6",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n = a_1 \\pm a_2 \\pm a_3 \\pm \\cdots \\pm a_k$ where $a_1, \\dots , a_k$ are simple.",
"... | {
"ground_truth": " k = 9 ",
"mean_reward": 0.28125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Let \\( r \\) represent the result of tripling both the base and exponent of \\( a^b \\), where \\( b \\) is non-zero. If \\( r \\) equals the product of \\( a^b \\) by \\( x^b \\), determine the value of \\( x \\).",
"role": "user"
}
] | {
"ground_truth": "27a^2",
"mean_reward": 0.359375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "Given the equation $3(4x + 5\\pi) = P$, simplify the expression $6(8x + 10\\pi)$ in terms of $P$. Express your answer as a coefficient of $P$.",
"role": "user"
}
] | {
"ground_truth": "4P",
"mean_reward": 0
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$. Sides $BC$ and $EF$ meet at $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$. Find the area of $ABHED$.\n[img]https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea246b3aa920d2464080a5.png[/... | {
"ground_truth": "54",
"mean_reward": 0.390625
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "A bottle contains 1001 blue balls, 1000 red balls, and 1000 green balls. At the same time, Xiaoming has enough balls of these three colors in his hand. Next, Xiaoming takes out two balls from the bottle each time, and then puts one or two balls back into the bottle according to the following rules... | {
"ground_truth": "red",
"mean_reward": 0.3125
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
[
{
"content": "8.2. Solve the inequality $\\sqrt{x^{2}-x-56}-\\sqrt{x^{2}-25 x+136}<8 \\sqrt{\\frac{x+7}{x-8}}$, and find the sum of its integer solutions that belong to the interval $[-25 ; 25]$.",
"role": "user"
}
] | {
"ground_truth": "-285",
"mean_reward": 0.375
} | AceReason-Math-Qwen3-4B-Instruct-eval-32k-64_partial | reasoning |
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