prompt listlengths 1 1 | data_source stringclasses 1
value | ability stringclasses 1
value | reward_model dict |
|---|---|---|---|
[
{
"content": "Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "62",
"mean_reward": 0.421875
} |
[
{
"content": "Consider an equilateral triangular grid $G$ with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points. Let $S$ be a closed... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "52 \\sqrt{3}",
"mean_reward": 0
} |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a function such that\n\\[ f(f(x) + y) = f(x^2 - y) + kf(x)y \\]\nfor all real numbers \\( x \\) and \\( y \\), and for some constant \\( k \\). Determine the number of possible values of \\( f(2) \\), and let \\( s \\) be the sum of all possible valu... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "8",
"mean_reward": 0.109375
} |
[
{
"content": "In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. The ma... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "150",
"mean_reward": 0
} |
[
{
"content": "The acute angle formed by the hands of a clock at 3:30 p.m. is \\( A^\\circ \\). Find \\( A \\).\n\nIf \\( \\tan (3A + 15)^\\circ = \\sqrt{B} \\), find \\( B \\).\n\nIf \\( \\log_{10} AB = C \\log_{10} 15 \\), find \\( C \\).\n\nThe points \\( (1, 3) \\), \\( (4, 9) \\), and \\( (2, D) \\) are col... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "5",
"mean_reward": 0.4375
} |
[
{
"content": "Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4",
"mean_reward": 0.015625
} |
[
{
"content": "How many natural numbers from 1 to 700, inclusive, contain the digit 7 at least once?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "133",
"mean_reward": 0
} |
[
{
"content": "In isosceles triangle $\\triangle ABC$ , $A$ is located at the origin and $B$ is located at $(20,0)$ . Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\\circ}$ . If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$ -ax... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "875",
"mean_reward": 0
} |
[
{
"content": "10. The sum $\\sum_{k=1}^{2020} k \\cos \\left(\\frac{4 k \\pi}{4041}\\right)$ can be written in the form $$ \\frac{a \\cos \\left(\\frac{p \\pi}{q}\\right)-b}{c \\sin ^{2}\\left(\\frac{p \\pi}{q}\\right)} $$ where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4049",
"mean_reward": 0.296875
} |
[
{
"content": "As shown in the figure, a large square is divided into six smaller squares. If two smaller squares share more than one common point, they are called adjacent. Fill the numbers $1, 2, 3, 4, 5, 6$ into the smaller squares in the right figure, with each smaller square containing one number such that ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "96",
"mean_reward": 0.109375
} |
[
{
"content": " $\\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more nu... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{2^n}{(n+1)!}",
"mean_reward": 0.25
} |
[
{
"content": "On the sides $\\mathrm{AB}$ and $\\mathrm{AC}$ of a right triangle $\\mathrm{ABC}\\left(\\angle B C A=90^{\\circ}\\right)$, right triangles $\\mathrm{ABT}$ and $\\mathrm{ACK}$ are constructed externally such that $\\angle A T B=\\angle A K C=90^{\\circ}$ and $\\angle A B T=\\angle A C K=60^{\\circ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "120",
"mean_reward": 0.421875
} |
[
{
"content": "$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \\cos \\angle B)(1 - \\cos \\angle ACE).$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "65",
"mean_reward": 0.234375
} |
[
{
"content": "A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "1",
"mean_reward": 0.1875
} |
[
{
"content": "Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (both Kelly and Jason know that $ n < 60$ , but that they don't know what the value of $ n$ is). The computer tells Kelly the unit digit of $ n$ , and it tells Jason the ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "10",
"mean_reward": 0.234375
} |
[
{
"content": "The medians from two vertices of a triangle are perpendicular. Show that the sum of the cotangent of the angles at those vertices is at least 2/3.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{2}{3}",
"mean_reward": 0.25
} |
[
{
"content": "Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. The original answer is in $\\frac{k}{m}$ format, where $\\frac{k}{m}$ is in simp... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "5",
"mean_reward": 0.078125
} |
[
{
"content": "Calculate the sum of the first six terms of the geometric series $3 + \\left(\\frac{1}{3}\\right) + \\left(\\frac{1}{3}\\right)^2 + \\left(\\frac{1}{3}\\right)^3 + \\left(\\frac{1}{3}\\right)^4 + \\left(\\frac{1}{3}\\right)^5$. Express your answer as a simplified fraction.",
"role": "user"
}... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{364}{81}",
"mean_reward": 0.390625
} |
[
{
"content": "In the figure, the triangle $\\triangle ABC$ is isosceles, with $\\angle BAC = 20^\\circ$.\n\nGiven that $BC = BD = BE$, determine the measure of the angle $\\angle BDE$.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "60",
"mean_reward": 0.0625
} |
[
{
"content": "Let $R = (10, 8)$. The lines whose equations are $9y = 18x$ and $15y = 6x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\\overline{PQ}$. Find the length of $PQ$ expressed as a reduced fraction $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "31",
"mean_reward": 0
} |
[
{
"content": "Brian starts at the point $\\left(1,0\\right)$ in the plane. Every second, he performs one of two moves: he can move from $\\left(a,b\\right)$ to $\\left(a-b,a+b\\right)$ or from $\\left(a,b\\right)$ to $\\left(2a-b,a+2b\\right)$. How many different paths can he take to end up at $\\left(28,-96\\r... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "70",
"mean_reward": 0.265625
} |
[
{
"content": "There are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Beverage, Food, Movie-Genre, Music-Genre, Pet, Sport.\nThe attributes have the following possible values:\n- Beverage: coffee, fanta, hot-chocolate, mirinda, sprite\n- F... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "hot-chocolate",
"mean_reward": 0.1875
} |
[
{
"content": "Let \\(ABCD\\) be a quadrilateral circumscribed about a circle with center \\(O\\). Let \\(O_1, O_2, O_3,\\) and \\(O_4\\) denote the circumcenters of \\(\\triangle AOB, \\triangle BOC, \\triangle COD,\\) and \\(\\triangle DOA\\). If \\(\\angle A = 120^\\circ\\), \\(\\angle B = 80^\\circ\\), and \... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "45",
"mean_reward": 0
} |
[
{
"content": "If \\(\\frac{18}{\\sqrt{x}} = 2\\), then the value of \\(x\\) is\n(A) 81\n(B) 36\n(C) 18\n(D) 9\n(E) 3",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "81",
"mean_reward": 0.25
} |
[
{
"content": "For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0$, $f(2) = 1$, and $f(4) = f(6) = 2$. Let $g(n)$ be the number of positive integers $k \\leq n$ such that $f(k) \\geq f(j)$ for all $j \\leq n$. Find $... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "136",
"mean_reward": 0.375
} |
[
{
"content": "For a nonempty set $\\, S \\,$ of integers, let $\\, \\sigma(S) \\,$ be the sum of the elements of $\\, S$ . Suppose that $\\, A = \\{a_1, a_2, \\ldots, a_{11} \\} \\,$ is a set of positive integers with $\\, a_1 < a_2 < \\cdots < a_{11} \\,$ and that, for each positive integer $\\, n\\... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "248",
"mean_reward": 0.0625
} |
[
{
"content": "Find all the solutions to\n\\[\\sqrt[3]{2 - x} + \\sqrt{x - 1} = 1.\\]Enter all the solutions, separated by commas.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "1,2,10",
"mean_reward": 0.4375
} |
[
{
"content": "There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "50",
"mean_reward": 0.40625
} |
[
{
"content": "Let \\( S = \\{ 1, 2, \\cdots, 2005 \\} \\). Find the smallest number \\( n \\) such that in any subset of \\( n \\) pairwise coprime numbers from \\( S \\), there is at least one prime number.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "16",
"mean_reward": 0.421875
} |
[
{
"content": "9.1. In the morning, a dandelion blooms, it flowers yellow for three days, on the fourth morning it turns white, and by the evening of the fifth day, it withers. On Monday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and on Wednesday - 15 yellow and 11 white. How many whi... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "6",
"mean_reward": 0.375
} |
[
{
"content": "Find all positive integers $ n $ with the following property: It is possible to fill a $ n \\times n $ chessboard with one of arrows $ \\uparrow, \\downarrow, \\leftarrow, \\rightarrow $ such that \n\n1. Start from any grid, if we follows the arrows, then we will eventually go back to the st... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "2",
"mean_reward": 0.25
} |
[
{
"content": "$ABCD$ is a trapezium such that $\\angle ADC=\\angle BCD=60^{\\circ}$ and $AB=BC=AD=\\frac{1}{2}CD$. If this trapezium is divided into $P$ equal portions $(P>1)$ and each portion is similar to trapezium $ABCD$ itself, find the minimum value of $P$.\n\nThe sum of tens and unit digits of $(P+1)^{200... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "34",
"mean_reward": 0.328125
} |
[
{
"content": "On the bases \\(AD\\) and \\(BC\\) of trapezoid \\(ABCD\\), squares \\(ADEF\\) and \\(BCGH\\) are constructed outside the trapezoid. The diagonals of the trapezoid intersect at point \\(O\\). Find the length of segment \\(AD\\) if \\(BC = 2\\), \\(GO = 7\\), and \\(GF = 18\\).",
"role": "user"... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{22}{7}",
"mean_reward": 0.296875
} |
[
{
"content": "Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)",
"r... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "17",
"mean_reward": 0
} |
[
{
"content": "Let $f(n)$ be the largest prime factor of $n^{2}+1$. Compute the least positive integer $n$ such that $f(f(n))=n$.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "89",
"mean_reward": 0.015625
} |
[
{
"content": "Prove that the section of pyramid $A B C D$ with a plane parallel to edges $A C$ and $B D$ is a parallelogram. Furthermore, show that for one such plane, this parallelogram is a rhombus. Find the side of this rhombus if $A C = a$ and $B D = b$.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{}{b}",
"mean_reward": 0
} |
[
{
"content": "In the trapezoid $ABCD$ with bases $AB$ and $CD$, let $M$ be the midpoint of side $DA$. If $BC=a$, $MC=b$ and $\\angle MCB=150^\\circ$, what is the area of trapezoid $ABCD$ as a function of $a$ and $b$?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\dfrac{ab}{2}",
"mean_reward": 0.046875
} |
[
{
"content": "B2. Graphically determine the intersection of the graphs of functions $f$ and $g$ with the rules $f(x)=3^{x}-1$ and $g(x)=x^{2}-2 x$. Verify the solution by calculation.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "(0, 0)",
"mean_reward": 0.140625
} |
[
{
"content": "Given the function \\( f(x)=\\left\\{\\begin{array}{l}2^{-x}-1, (x \\leq 0), \\\\ f(x-1), (x>0),\\end{array}\\right. \\) if the equation \\( f(x)=x+a \\) has exactly two distinct real roots, then the range of values for the real number \\( a \\) is $\\qquad$ .",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "(-\\infty,1)",
"mean_reward": 0.09375
} |
[
{
"content": "Scatterbrained Scientist walks to work every morning and back home every evening. He has two umbrellas and takes one with him under two conditions: it is raining outside or there is no umbrella at his destination. Over time, he calculates that he carries an umbrella 20% of the time. Find the proba... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{1}{9}",
"mean_reward": 0.125
} |
[
{
"content": "Let $X$ be a 2024 digit perfect square. Let $a(X)$ be the 1012 digit number formed from reading the\nfirst 1012 digits of $X$, in order, and let $b(X)$ be the 1012 digit number formed from reading the last\n1012 digits of $X$, in order. Given that $X$ is the unique choice that maximizes $a(X) – b(... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "9108",
"mean_reward": 0.375
} |
[
{
"content": "What is the minimum value of the expression \n\n\\[\n\\sqrt{x^{2}+y^{2}-2x-2y+2} + \\sqrt{x^{2}+y^{2}-2x+4y+2\\sqrt{3}y+8+4\\sqrt{3}} + \\sqrt{x^{2}+y^{2}+8x+4\\sqrt{3}x-4y+32+16\\sqrt{3}}\n\\]\n\nfor \\( (x, y) \\in \\mathbb{R} \\)?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "3\\sqrt{6}+4\\sqrt{2}",
"mean_reward": 0.15625
} |
[
{
"content": "Find the number of positive integers $n \\le 1200$ such that $14n$ is a perfect square.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "9",
"mean_reward": 0.421875
} |
[
{
"content": "Given two positive integers $m,n$, we say that a function $f : [0,m] \\to \\mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties:\n\ni) $f$ is continuous;\nii) $f(0) = 0$, $f(m) = n$;\niii) If $t_1, t_2\\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\\in \\mathbb{Z}$ and $f(... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\gcd(m, n) = 1",
"mean_reward": 0.125
} |
[
{
"content": "On a $3 \\times 3$ chessboard, each square contains a knight with $\\frac{1}{2}$ probability. What is the probability that there are two knights that can attack each other? (In chess, a knight can attack any piece which is two squares away from it in a particular direction and one square away in a... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{209}{256}",
"mean_reward": 0.140625
} |
[
{
"content": "Mr. Johnson is a retired teacher who actively attends union meetings to support his fellow educators. Last month, he attended 5 different union meetings. At each meeting, he brought along 3 retired friends to also show their support. If each meeting lasted 2 hours, and they spent an additional 30 ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "50",
"mean_reward": 0.234375
} |
[
{
"content": "The Hangzhou Asian Games are underway, and table tennis, known as China's \"national sport,\" is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{3}{4}",
"mean_reward": 0.328125
} |
[
{
"content": "## Task B-1.1.\n\nHow many ordered pairs of natural numbers $(a, b)$ are there such that $a+b=1000$ and neither of the numbers $a$ and $b$ contains the digit 0 in their decimal representation?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "738",
"mean_reward": 0.28125
} |
[
{
"content": "5. Given the ellipse $\\frac{x^{2}}{1}+y^{2}=1$ with left and right foci $F_{1}, F_{2}$, point $P$ is on the line $l: x+2 \\sqrt{3} y-4 \\sqrt{3}=0$. When $\\angle F_{1} P F_{2}$ takes the maximum value,\n$\\frac{\\left|P F_{1}\\right|}{\\left|P F_{2}\\right|}$ is",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\dfrac{\\sqrt{15}}{3}",
"mean_reward": 0.453125
} |
[
{
"content": "1. (3 points) The left figure below is the recently discovered Archimedes' \"Stomachion\" puzzle, dividing a square into 14 polygonal pieces: After research, experts found that these 14 pieces can be correctly drawn on a square with a side length of $12 \\mathrm{~cm}$, as shown in the figure. Ques... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "12",
"mean_reward": 0.171875
} |
[
{
"content": "For how many of the given drawings can the six dots be labelled to represent the links between suspects?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "2",
"mean_reward": 0.171875
} |
[
{
"content": "Alex, a busy corporate lawyer, works 60 hours each week and wants to find time for a meaningful relationship. Alex decides to allocate some hours from their workweek to spend with a potential partner. If Alex manages to reduce their work hours by 10% each week, how many hours will Alex have availa... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "6",
"mean_reward": 0.484375
} |
[
{
"content": "\nProblem 3. Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \\ldots, x_{n}$ (not necessarily different), with $1 \\leq x_{k} \\leq n, 1 \\leq k \\leq n$, and such that\n\n$$\nx_{1}+x_{2}+\\cdots+x_{n}=\\frac{n(n+1)}{2}, \\quad \\text { and } \\quad x_{1} ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "9",
"mean_reward": 0
} |
[
{
"content": "In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types?",
"role"... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "69",
"mean_reward": 0.4375
} |
[
{
"content": "## 12. Figure\n\nOn a plane, there are five points such that no three of them lie on the same line. The points are connected by straight lines. What is the number of intersection points of the lines (excluding the original five points), if none of the drawn lines are parallel to each other?",
... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "15",
"mean_reward": 0.375
} |
[
{
"content": "In triangle \\(ABC\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\). A circle is drawn through points \\(B\\), \\(C\\), and point \\(D\\), which lies on side \\(AB\\), intersecting side \\(AC\\) at point \\(E\\). Find \\(AE\\) if \\(AD = 3\\), \\(BD = 1\\), and \\(EC = 4\\). Find the radius of... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\sqrt{7}",
"mean_reward": 0.078125
} |
[
{
"content": "Let \\( n \\) be a positive integer. A pair of \\( n \\)-tuples \\((a_{1}, \\ldots, a_{n}) \\) and \\((b_{1}, \\ldots, b_{n}) \\) with integer entries is called an exquisite pair if\n\\[ \\left| a_{1} b_{1} + \\cdots + a_{n} b_{n} \\right| \\leq 1. \\]\nDetermine the maximum number of distinct \\(... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "n^2+n+1",
"mean_reward": 0.0625
} |
[
{
"content": "Can the 8 numbers \\(1, 2, \\cdots, 8\\) be arranged on the vertices of a regular octagon such that the sum of the numbers at any three consecutive vertices is:\n(1) Greater than 11?\n(2) Greater than 13?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "No",
"mean_reward": 0.015625
} |
[
{
"content": "\\left.\\begin{array}{l}{[\\quad \\text { Law of Sines }} \\\\ {\\left[\\begin{array}{l}\\text { Angle between a tangent and a chord }]\\end{array}\\right]}\\end{array}\\right]\n\nIn triangle $ABC$, angle $\\angle B$ is equal to $\\frac{\\pi}{6}$. A circle with a radius of 2 cm is drawn through po... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\sqrt{6}",
"mean_reward": 0.359375
} |
[
{
"content": "Alice drew a regular 2021-gon in the plane. Bob then labelled each vertex of the 2021-gon with a real number, in such a way that the labels of consecutive vertices differ by at most 1. Then, for every pair of non-consecutive vertices whose labels differ by at most 1, Alice drew a diagonal connecti... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "2018",
"mean_reward": 0
} |
[
{
"content": "Find the last two digits of $\\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$).",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "20",
"mean_reward": 0.21875
} |
[
{
"content": "When askes: \"What time is it?\", father said to a son: \"Quarter of time that passed and half of the remaining time gives the exact time\". What time was it?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "9:36",
"mean_reward": 0.234375
} |
[
{
"content": "4. Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_{1}, x_{2}, \\ldots, x_{n}$. Find all solutions for each such $n$.\n$$\n\\begin{aligned}\nx_{1}+x_{2}+\\cdots+x_{n} & =16 \\\\\n\\frac{1}{x_{1}}+\\frac{1}{x_{2}}+\... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4",
"mean_reward": 0.3125
} |
[
{
"content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`, `Eric`\n... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "xiaomi mi 11",
"mean_reward": 0.484375
} |
[
{
"content": "9.5. At the summit, 2018 politicians gathered. Every two of them were supposed to hold private negotiations. At some point, it was found that among any four, there is one who has already talked to the other three. What is the maximum number of negotiations left to be held?",
"role": "user"
}... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "3",
"mean_reward": 0.453125
} |
[
{
"content": "Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly eight steps to become palindromes?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "187",
"mean_reward": 0.015625
} |
[
{
"content": "Given real numbers \\( a_{0}, a_{1}, a_{2}, \\cdots, a_{n} \\) satisfying \\( a_{n}=0 \\) and\n\\[ a_{k}=c+\\sum_{i=k}^{n-1} a_{i-k}(a_{i}+a_{i+1}), \\quad k=0,1,2, \\cdots, n-1, \\]\nprove that \\( c \\leq \\frac{1}{4n} \\).",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\le\\frac{1}{4n}",
"mean_reward": 0
} |
[
{
"content": "# 7. Clone 1\n\nIn the expression OL $*$ IM $* P *$ IA * DA, it is required to replace the asterisks with two plus signs and two minus signs, and to replace the letters with digits according to the rules of a cryptarithm (identical letters with identical digits, and different letters with differen... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "263",
"mean_reward": 0.140625
} |
[
{
"content": "In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "54",
"mean_reward": 0.1875
} |
[
{
"content": "The minimum possible sum of the three dimensions of a rectangular box with a volume of $1729$ in$^3$.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "39",
"mean_reward": 0.0625
} |
[
{
"content": "Example 13. Given that the characteristic of $\\lg x^{\\frac{3}{2}}$ is the solution to the inequality\n$$\na^{2}-8 a+15<0\n$$\n\nand the mantissa is the sum of the solutions to the equation\n$$\n\\frac{1}{4}\\left|y+\\frac{1}{2}\\right|+\\frac{6}{121} \\sqrt{z-1}=0\n$$\n\nfind the value.",
"r... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "1000",
"mean_reward": 0.484375
} |
[
{
"content": "Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "7",
"mean_reward": 0.03125
} |
[
{
"content": "33. How many people got lost in the mountains?\n\nAlice and the Griffin had to wait several minutes before the Tortoise Quasi gathered his strength and could continue.\n\n- You see,一began the Tortoise Quasi.\n- I don't see anything!- the Griffin cut in.\n\nThe Tortoise Quasi did not respond, only ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "3",
"mean_reward": 0
} |
[
{
"content": "There are $n$ teams that need to hold a double round-robin tournament (each pair of teams competes twice, with each team having one match at their home field). Each team can play multiple away games in a week (from Sunday to Saturday). However, if a team has a home game in a particular week, they ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "6",
"mean_reward": 0.015625
} |
[
{
"content": "Circles $\\mathcal{P}$ and $\\mathcal{Q}$ have radii $1$ and $4$ , respectively, and are externally tangent at point $A$ . Point $B$ is on $\\mathcal{P}$ and point $C$ is on $\\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\\ell$ through... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "129",
"mean_reward": 0.5
} |
[
{
"content": "Ant Borya moves on the coordinate plane, starting from the point \\( P_{0} = (0, 0) \\), moving to the point \\( P_{1} = (1, 0) \\), and then spiraling counterclockwise. Points with integer coordinates, which he reaches, form a sequence \\( P_{n} \\). Find the coordinates of the point \\( P_{1557}... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "(20, 17)",
"mean_reward": 0.0625
} |
[
{
"content": "Using an electric stove with a power of $P=500 \\mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\\Delta T=2^{\\circ} \\mathrm{C}$, and after turning off the heater, the temperature decreases to the ini... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "2.38",
"mean_reward": 0.28125
} |
[
{
"content": "Line segment $\\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, wha... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "50",
"mean_reward": 0.203125
} |
[
{
"content": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`\n - People use un... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "samsung galaxy s21",
"mean_reward": 0.171875
} |
[
{
"content": "Jamie paid for a $2.40 meal using exactly 50 coins consisting of pennies, nickels, and dimes. How many dimes did Jamie use if no change was received?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "10",
"mean_reward": 0.328125
} |
[
{
"content": "The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3,0)--(3,1));\ndraw(... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "-7",
"mean_reward": 0
} |
[
{
"content": "7. [4] My friend and I are playing a game with the following rules: If one of us says an integer $n$, the opponent then says an integer of their choice between $2 n$ and $3 n$, inclusive. Whoever first says 2007 or greater loses the game, and their opponent wins. I must begin the game by saying a ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "6",
"mean_reward": 0.0625
} |
[
{
"content": "Let $\\{a_n\\}$ be an arithmetic sequence with a non-zero common difference, and all terms are positive. Evaluate the inequality $a_1 \\cdot a_8$ compared to $a_4 \\cdot a_5$. Express your answer as a specific inequality (<, >, or =).",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "<",
"mean_reward": 0.234375
} |
[
{
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "72",
"mean_reward": 0
} |
[
{
"content": "Find the minimum value of the real number \\(\\lambda\\), such that for any integer \\(n \\geq 2\\) and positive real numbers \\(a_{1}, a_{2}, \\cdots, a_{n}\\) satisfying \\(\\sum_{i=1}^{n} a_{i}=n\\), the following inequality always holds:\n$$\n\\sum_{i=1}^{n} \\frac{1}{a_{i}}-\\lambda \\prod_{i... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "e",
"mean_reward": 0.359375
} |
[
{
"content": "Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "-2",
"mean_reward": 0.015625
} |
[
{
"content": "The function $f(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $f(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $f(x) \\le f(y).$\n(iii) $f(1 - x) = 1 - f(x)$ for all $0 \\le x \\le 1.$\n(iv) $f \\left( \\frac{x}{3} \\right) = \\frac{f(x)}{2}$ for $0 \\le x \\le 1.$\n\nFi... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "11",
"mean_reward": 0.203125
} |
[
{
"content": "Rebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistance of \\(a\\) and \\(b\\) ohms respectively, and combines them into one by one of the following methods:\n- Connect them in series, which produces a resistor with resistance of \\(a+b\\... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "15",
"mean_reward": 0.453125
} |
[
{
"content": "2. Can 2010 be written as the sum of squares of $k$ distinct prime numbers? If so, find the maximum value of $k$; if not, please briefly explain the reason.",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "7",
"mean_reward": 0.28125
} |
[
{
"content": "35. Is there a natural number $n$ such that $n^{n}+(n+1)^{n}$ is divisible by 1987?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "993",
"mean_reward": 0
} |
[
{
"content": "Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$ , respectively, with $FA=5$ and $CD=2$ . Point $E$ lies on side $CA$ such that $\\angle DEF = 60^\\circ$ . The area of triangle $DEF$ is $14\\sqrt{3}$ . The two possible values of the ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "989",
"mean_reward": 0.21875
} |
[
{
"content": "What is the minimum total number of boxes that Carley could have bought if each treat bag contains exactly 1 chocolate, 1 mint, and 1 caramel, and chocolates come in boxes of 50, mints in boxes of 40, and caramels in boxes of 25?",
"role": "user"
}
] | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "17",
"mean_reward": 0.140625
} |
[
{
"content": "Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition:\nThe difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive int... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4040100",
"mean_reward": 0
} |
[
{
"content": "1. [3] Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_{0}, F_{1}, F_{2}, F_{3}, \\ldots$ of figures as follows:\n- $F_{0}$ is a square.\n- For every positive integer $i, F_{i}$ is randomly chosen... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "7",
"mean_reward": 0.484375
} |
[
{
"content": "Let \\( a_{0}, a_{1}, a_{2}, \\ldots \\) be a sequence of real numbers such that \\( a_{0} = 0 \\), \\( a_{1} = 1 \\), and for every \\( n \\geqslant 2 \\) there exists \\( 1 \\leqslant k \\leqslant n \\) satisfying\n\\[ a_{n} = \\frac{a_{n-1} + \\cdots + a_{n-k}}{k} . \\]\n\nFind the maximal poss... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{2016}{2017^2}",
"mean_reward": 0
} |
[
{
"content": "In acute $\\triangle ABC$ with centroid \\( G \\), \\( AB = 22 \\) and \\( AC = 19 \\). Let \\( E \\) and \\( F \\) be the feet of the altitudes from \\( B \\) and \\( C \\) to \\( AC \\) and \\( AB \\) respectively. Let \\( G' \\) be the reflection of \\( G \\) over \\( BC \\). If \\( E, F, G \\)... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "13",
"mean_reward": 0.03125
} |
[
{
"content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Bob`, `Carol`, `Alice`, `Peter`, `Arnold`, `Eric`\n... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4",
"mean_reward": 0.328125
} |
[
{
"content": "\nA city has $n$ bus lines satisfying the following conditions:\n(1) Any bus stop is serviced by at most 3 bus lines;\n(2) Any bus line has at least two stops;\n(3) For any two specific bus lines, there is a third line such that passengers can transfer from either of the two specified lines to the... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{5}{6}(n-5)",
"mean_reward": 0.46875
} |
[
{
"content": "In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(2,... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "\\frac{1}{2}",
"mean_reward": 0
} |
[
{
"content": "Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron where each edge is 1 meter. A bug starts at vertex $A$ and at each vertex chooses randomly among the three incident edges to move along. Compute the probability $p$ that the bug returns to vertex $A$ after exactly 10 meters, where $p ... | Polaris-AceReason-Math-combined | reasoning | {
"ground_truth": "4921",
"mean_reward": 0
} |
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