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[
{
"role": "user",
"content": "16. Lego Explorer Guguji arrives at a medieval kingdom, which is ruled by an evil wizard. With the help of Guguji and his friends, the prince defeats the wizard and returns to his castle. The castle is built with many cubes with an edge length of 1 meter, arranged from top to b... | 4901 | polaris53k | 36,794 | 0/8 | null | zero | 0 | 0 | 17,380 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 1925 | 2 | confirmed | true |
[
{
"role": "user",
"content": "8. In the figure shown in Figure 3, on both sides of square $P$, there are $a$ and $b$ squares to the left and right, and $c$ and $d$ squares above and below, where $a$, $b$, $c$, and $d$ are positive integers, satisfying\n$$\n(a-b)(c-d)=0 \\text {. }\n$$\n\nThe shape formed by... | 13483236 | polaris53k | 46,444 | 1/8 | null | mixed1to3 | 2 | 0.5 | 15,712 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | 15\,116\,400 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Construct a \\( \\varepsilon - \\delta \\) proof to show that \\( \\lim_{x \\to 27} 2x^{2/3} = 18 \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 18 | deepmath103k | 28,392 | 4.0 | Mathematics -> Precalculus -> Limits | mixed1to3 | 2 | 0.5 | 20,621 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | \delta = \min\left\{27,\ \frac{9\varepsilon}{14}\right\} \text{ works: } 0<|x-27|<\delta \implies |2x^{2/3}-18|<\varepsilon | 1 | confirmed | true |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \\frac{p(x)}{x(1-x)}$ . If $q(x) = q\\left... | 521 | dapo17k | 7,361 | null | null | mixed1to3 | 3 | 0.75 | 18,592 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Let $U \\subset W \\subset V$ be three linear spaces with dimensions 3, 6, and 10, respectively. Define $E \\subset L(V,V)$ as the set of linear transformations $f: V \\rightarrow V$ such that $f(U) \\subset U$ and $f(W) \\subset W$. Determine the dimension of $E$. \\( \\dim E ... | 67 | deepmath103k | 24,282 | 6.0 | Mathematics -> Algebra -> Linear Algebra -> Linear Transformations | mixed1to3 | 3 | 0.75 | 13,162 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个不均匀的骰子,掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列。独立地先后郑该骰子两次,所得的点数分别记为 $a, b$ 。若事件 \" $a+b=7$ \" 发生的概率为 $\\frac{1}{7}... | 25 | dapo17k | 12,522 | null | null | mixed1to3 | 3 | 0.75 | 9,569 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int\\limits_{0}^{\\infty} {\\frac{2}{3x^{4/3}} \\sqrt{\\frac{\\lambda}{2\\pi}} \\exp \\left( \\frac{-\\lambda (x^{2/3}-\\mu)^2}{2\\mu^2 x^{2/3}} \\right) dx} \\) for \\(\\mu>0, \\lambda>0\\).\n\nPlease reason step by step, and put your final answer ... | 1 | deepmath103k | 33,781 | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | mixed1to3 | 2 | 0.5 | 12,381 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | \mu | 1 | confirmed | true |
[
{
"role": "user",
"content": "Determine the values of \\( s \\) for which the integral \\( \\int_{\\mathbb{R}^n}\\left(\\frac{1}{1+|x|^2}\\right)^{s}dx \\) converges.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | s > \dfrac{n}{2} | deepmath103k | 14,902 | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | mixed1to3 | 3 | 0.75 | 6,925 | 0 | false | false | 4 | 4 | 4 | 1 | 3 | 0 | confirmed | false | |
[
{
"role": "user",
"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: `Carol`, `Peter`, `Alice`, `Eric... | Kailyn | polaris53k | 41,999 | 0/8 | null | mixed1to3 | 2 | 0.5 | 20,806 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "26th CanMO 1994 Problem 4 AB is the diameter of a circle. C is a point not on the line AB. The line AC cuts the circle again at X and the line BC cuts the circle again at Y. Find cos ACB in terms of CX/CA and CY/CB. Solution\n\nPlease reason step by step, and put your final ans... | \sqrt{ \dfrac{CX}{CA} \cdot \dfrac{CY}{CB} } | polaris53k | 51,168 | 0/8 | null | mixed1to3 | 1 | 0.25 | 27,633 | 2 | false | false | 4 | 4 | 2 | 2 | 0 | \cos\angle ACB \;=\; \pm\sqrt{\frac{CX}{CA}\cdot\frac{CY}{CB}}\; | 2 | confirmed | true |
[
{
"role": "user",
"content": "Determine the smallest integer $n$ for which the number of pairs $(x, y)$ satisfying $x \\ge y$, $\\operatorname{lcm}(x, y) = n!$, and $\\gcd(x, y) = 2023$ is greater than 2023. Note that $2023 = 7 \\times 17^2$. What is this value of $n$?\n\nPlease reason step by step, and put... | 41 | deepmath103k | 39,451 | 7.0 | Mathematics -> Number Theory -> Factorization | mixed1to3 | 2 | 0.5 | 11,756 | 0 | false | false | 4 | 4 | 1 | 1 | 0 | 37 | 2 | uncertain | false |
[
{
"role": "user",
"content": "Evaluate the limit of the integral: $$\\lim_{n \\rightarrow \\infty}\\int_{0}^{\\pi/4}{\\frac {\\sin^2(\\frac{nx}{2})}{n\\,\\sin^2(\\frac{x}{2})} }\\,dx$$\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | \pi | deepmath103k | 74,866 | 7.0 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | mixed1to3 | 3 | 0.75 | 10,244 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Let's call a word any finite sequence of letters of the Russian alphabet. How many different four-letter words can be made from the letters of the word КАША? And from the letters of the word ХЛЕБ? Indicate the sum of the found numbers in the answer.\n\nPlease reason step by ste... | 36 | polaris53k | 11,795 | 4/8 | null | mixed1to3 | 3 | 0.75 | 6,267 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "In the diagram, \\(C\\) lies on \\(AE\\) and \\(AB=BC=CD\\). If \\(\\angle CDE=t^{\\circ}, \\angle DEC=(2t)^{\\circ}\\), and \\(\\angle BCA=\\angle BCD=x^{\\circ}\\), determine the measure of \\(\\angle ABC\\).\n\nPlease reason step by step, and put your final answer within \\b... | 60 | polaris53k | 7,298 | 2/8 | null | mixed1to3 | 3 | 0.75 | 18,487 | 0 | false | false | 4 | 2 | 0 | 0 | 0 | \angle ABC = 180 - 3t | 1 | uncertain | false |
[
{
"role": "user",
"content": "A group of 6 students decided to make *study groups* and *service activity groups* according to the following principle:\n\nEach group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the stude... | 8 | polaris53k | 50,084 | 0/8 | null | mixed1to3 | 1 | 0.25 | 15,928 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Does the quotient of a complex algebraic group by its neutral component always split? Specifically, let $G$ be a complex algebraic group and $G^\\circ \\subseteq G$ be the connected component containing the neutral element. Given that $G^\\circ$ is a normal subgroup and $G/G^\\... | Yes | deepmath103k | 21,938 | 8.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | mixed1to3 | 3 | 0.75 | 9,021 | 0 | false | false | 4 | 4 | 0 | 0 | 0 | \text{No. The group } G = N_{\mathrm{SL}_2(\mathbb{C})}(T) \text{ has } G^\circ \cong \mathbb{C}^*,\ G/G^\circ \cong \mathbb{Z}/2, \text{ but no subgroup } Q \cong G/G^\circ \text{ complements } G^\circ: \text{ every element of the nontrivial coset squares to } -I \in G^\circ. | 1 | uncertain | false |
[
{
"role": "user",
"content": "Let $A$ and $B$ be compact metric spaces with Borel probability measures $m_A$ and $m_B$, respectively. Suppose $f: A \\to B$ is a continuous surjection. If $m_A(K) = 1$ for a Borel set $K \\subseteq A$, is it true that $m_B(f(K)) = 1$? Assume $A$ is second-countable.\n\nPlease... | No | deepmath103k | 30,153 | 8.5 | Mathematics -> Other | mixed1to3 | 3 | 0.75 | 9,912 | 0 | false | false | 4 | 4 | 2 | 0 | 2 | \text{Yes, } m_B(f(K)) = 1 \text{ (given } m_B = f_\ast m_A\text{); measurability of } f(K) \text{ holds since } f(K) \text{ is analytic.} | 1 | confirmed | false |
[
{
"role": "user",
"content": "3. In a circle with radius $R$, determine the central angle corresponding to a circular segment with the following property: the chord of the segment is equal to the perimeter of the circle with the largest radius that is inscribed in it.\n\nPlease reason step by step, and put ... | 4 \arctan\left(\frac{2}{\pi}\right) | polaris53k | 37,678 | 1/8 | null | mixed1to3 | 2 | 0.5 | 10,393 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Evaluate the limit of the double integral: \\[ \\lim_{b \\to 0^+} \\int_{b}^1 \\int_b^1 \\frac{y-x}{(y+x)^3} \\, dy \\, dx \\]\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 0 | deepmath103k | 32,586 | 7.0 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | mixed1to3 | 3 | 0.75 | 16,464 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stad... | 7 | dapo17k | 6,961 | null | null | mixed1to3 | 3 | 0.75 | 2,169 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Find the integer solution to the equation \\(975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + 195j + 125k = 1002\\) that minimizes the sum \\(a + b + c + d + e + f + g + h + i + j + k\\), given the constraints:\n\n\\[0 \\leq a \\leq 27\\]\n\\[0 \\leq b \\leq 37\\]... | 3 | deepmath103k | 40,638 | 5.5 | Mathematics -> Other | zero | 0 | 0 | 17,549 | 0 | false | false | 4 | 4 | 4 | 2 | 2 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`,... | prince | polaris53k | 35,872 | 0/8 | null | mixed1to3 | 1 | 0.25 | 22,238 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of th... | 2007 | dapo17k | 5,338 | null | null | mixed1to3 | 1 | 0.25 | 17,477 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \\frac{\\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.\n\n(Ⅰ) Find the equation of the ellipse $C$.\n\n(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse ... | \frac{\sqrt{2}}{2} | polaris53k | 16,333 | 2/8 | null | mixed1to3 | 3 | 0.75 | 19,175 | 0 | false | false | 4 | 4 | 4 | 0 | 4 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "In a finite semigroup with a tail and a cycle, determine the number of possible neutral elements in the cycle.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 1 | deepmath103k | 51,736 | 6.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | mixed1to3 | 3 | 0.75 | 11,040 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的质数 $p$, 满足 $(p, N)=1$,其中, $N$ 为所有满足以下条件的 $\\left(a_{0}, a_{1}, \\cdots\\right.$, $a_{2012}$ )的个数:\n(1)... | 67 | dapo17k | 11,052 | null | null | mixed1to3 | 1 | 0.25 | 17,307 | 1 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Evaluate the triple integral: \\[ \\int_0^1 \\int_0^{z\\pi} \\int_{\\frac{y}{\\pi}}^z \\sin\\frac{y}{x} \\, dx \\, dy \\, dz \\]\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | \dfrac{1}{3} | deepmath103k | 32,881 | 7.0 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | mixed1to3 | 3 | 0.75 | 7,054 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of the prime factors of $67208001$, given that $23$ is one.\n\nRemember to put your answer on... | 781 | dapo17k | 3,517 | null | null | mixed1to3 | 3 | 0.75 | 15,794 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"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: `Alice`, `Eric`, `Peter`, `Carol... | tesla model 3 | polaris53k | 42,062 | 0/8 | null | mixed1to3 | 2 | 0.5 | 24,368 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"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: `Eric`, `Arnold`, `Carol`, `Bob`... | 4 | polaris53k | 44,049 | 0/8 | null | mixed1to3 | 3 | 0.75 | 14,478 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "A finite increasing sequence \\(a_{1}, a_{2}, \\ldots, a_{n}\\) of natural numbers is given, where \\(n \\geq 3\\), and for all \\(k \\leq n-2\\) the following equality holds: \\(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\\). The sequence must include \\(a_{k} = 2022\\). Determine the max... | 225 | polaris53k | 20,134 | 0/8 | null | mixed1to3 | 3 | 0.75 | 21,312 | 1 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that t... | 244 | dapo17k | 15,108 | null | null | mixed1to3 | 3 | 0.75 | 9,710 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "A2 From each of the eight vertices of a cube, a piece is sawn off so that instead of a vertex, a triangular face with three vertices appears. Each pair of the 24 vertices of the resulting body is connected by a line segment.\n\nHow many of these line segments lie, except for th... | 120 | polaris53k | 47,089 | 5/8 | null | mixed1to3 | 2 | 0.5 | 11,159 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the secon... | 26 | polaris53k | 29,120 | 0/8 | null | mixed1to3 | 3 | 0.75 | 20,292 | 1 | false | false | 4 | 4 | 3 | 3 | 0 | 20 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Determine the dimension of the vector space $V_r$ over the complex numbers $\\mathbb{C}$, where $V_r$ consists of all holomorphic functions $f: \\mathbb{C} \\rightarrow \\mathbb{C}$ such that $|f(z)| < \\infty$ for all $z \\in \\mathbb{C}$. Provide your answer as a single numbe... | \infty | deepmath103k | 40,636 | 7.5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | zero | 0 | 0 | 2,322 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Find the largest possible set \\( A \\) consisting of 2-digit natural numbers (from 01 to 99) such that every subset of \\( A \\) has a unique sum. What is the size of this set?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 7 | deepmath103k | 56,991 | 7.0 | Mathematics -> Discrete Mathematics -> Combinatorics | mixed1to3 | 2 | 0.5 | 23,102 | 0 | false | false | 4 | 1 | 1 | 1 | 0 | 0 | uncertain | false | |
[
{
"role": "user",
"content": "## Zadatak B-4.3.\n\nPrvi, peti i jedanaesti član rastućeg aritmetičkog niza istovremeno su tri uzastopna člana geometrijskog niza. Odredite sto dvanaesti član aritmetičkog niza, ako je njegov prvi član jednak 136 .\n\n\nPlease reason step by step, and put your final answer wit... | 2023 | polaris53k | 44,517 | 0/8 | null | zero | 0 | 0 | 3,397 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "The number \\(27,000,001\\) has exactly four prime factors. Find their sum.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 652 | polaris53k | 15,665 | 4/8 | null | mixed1to3 | 3 | 0.75 | 16,536 | 1 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Find the value of \\(x\\) such that \\(\\int_{0}^{x}\\{t\\}dt = \\int_{0}^{\\{x\\}}tdt\\), where \\(x > 0\\), \\(x \\notin \\mathbb{Z}\\), and \\(\\{x\\}\\) is the fractional part of \\(x\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | (0,1) | deepmath103k | 26,468 | 5.0 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | mixed1to3 | 2 | 0.5 | 7,813 | 0 | false | false | 4 | 4 | 4 | 1 | 3 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正方体中随机取 3 条棱, 它们两两异面的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式,其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your a... | 57 | dapo17k | 12,771 | null | null | mixed1to3 | 3 | 0.75 | 14,000 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 63 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter i... | 86 | dapo17k | 8,269 | null | null | mixed1to3 | 2 | 0.5 | 10,573 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 102 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Numbers 1 through 9 are written in the cells of a $3 \\times 3$ table. All possible sums of the numbers in adjacent (by side) cells are then listed in a notebook. What is the minimum number of different sums that could be listed in the notebook?\n\nPlease reason step by step, a... | 4 | polaris53k | 33,675 | 0/8 | null | mixed1to3 | 2 | 0.5 | 23,572 | 0 | false | false | 4 | 3 | 3 | 3 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Let $X_1, \\ldots, X_n$ be independent random variables where $X_i \\sim N(\\beta + \\gamma z_i, \\sigma^2)$, with $\\sum_{i=1}^n z_i = 0$ and $\\sum_{i=1}^n z_i^2 = 1$. Define $Y = \\frac{1}{n} \\sum_{i=1}^n X_i$ and $Z = \\sum_{i=1}^n z_i X_i$. Calculate the correlation coeff... | 0 | deepmath103k | 75,590 | 7.0 | Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics | mixed1to3 | 3 | 0.75 | 16,441 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"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: `Peter`, `Eric`, `Arnold`, `Bob`... | 4 | polaris53k | 53,148 | 0/8 | null | mixed1to3 | 2 | 0.5 | 17,069 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Evaluate the curvilinear integral \\(\\int_{\\gamma} x \\, ds\\) where the curve \\(\\gamma\\) is defined by the polar equation \\(\\rho = 1 + \\cos(\\theta)\\) for \\(\\theta \\in [-\\pi, \\pi]\\). The curve is parametrized by\n\\[ \\begin{cases} x = \\rho(\\theta)\\cos(\\thet... | \dfrac{32}{5} | deepmath103k | 57,600 | 7.0 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | mixed1to3 | 3 | 0.75 | 13,928 | 0 | false | false | 4 | 4 | 4 | 3 | 1 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "An $n \\times n$ table is filled with real numbers such that the elements decrease monotonically in every row and every column. We want to decide whether a number $a$ is in the table. For this, we can compare $a$ with any element of the table. How many comparisons are needed to... | 2n - 1 | polaris53k | 47,105 | 2/8 | null | mixed1to3 | 3 | 0.75 | 16,281 | 0 | false | false | 4 | 2 | 2 | 2 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Baba Yaga in her hut on chicken legs has kept fairy tale animals. All of them, except for two, are Talking Cats; all, except for two, are Wise Owls; the rest are Whiskered Cockroaches. How many inhabitants are there in Baba Yaga's hut?\n\nPlease reason step by step, and put you... | 3 | polaris53k | 48,178 | 2/8 | null | mixed1to3 | 3 | 0.75 | 12,402 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "There are 5 houses, numbered 1 to 5 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: `Peter`, `Eric`, `Arnold`, `Bob`... | Peter | polaris53k | 41,972 | 0/8 | null | mixed1to3 | 3 | 0.75 | 12,711 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Given the sequence \\( 0 \\leqslant a_{k} \\leqslant 1 \\) for \\( k = 1, 2, \\cdots, 2002 \\), with \\( a_{2003} = a_{1} \\) and \\( a_{2004} = a_{2} \\), determine the maximum value of the expression \\( \\sum_{k=1}^{20002} \\left( a_{k} - a_{k+1} a_{k+2} \\right) \\).\n\nPle... | 1001 | deepmath103k | 92,546 | 7.0 | Mathematics -> Algebra -> Intermediate Algebra -> Other | zero | 0 | 0 | 17,745 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Determine the coefficient of degree \\(-1\\) in the formal differential form \\(\\frac{df}{f}\\) for a formal Laurent power series \\(f=\\sum_{i\\ge m} a^i t^i\\) where \\(m\\in \\mathbb{Z}\\). Can you derive an explicit expression for this coefficient?\n\nPlease reason step by... | m | deepmath103k | 55,927 | 6.5 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | mixed1to3 | 3 | 0.75 | 12,898 | 0 | false | false | 4 | 4 | 4 | 1 | 3 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "One, (20 points) A research study group from Class 1, Grade 3 of Yuhong Middle School conducted a survey on students' lunch time at the school canteen. It was found that within a unit of time, the number of people buying lunch at each window and the number of people choosing to... | 5 | polaris53k | 44,431 | 4/8 | null | zero | 0 | 0 | 18,571 | 0 | false | false | 4 | 3 | 3 | 3 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "A convex quadrilateral has diagonals that are perpendicular to each other. Consider the sum of the diameters of the 4 circles inscribed in the parts established by these diagonals. How can this sum be expressed using the diagonals and the perimeter of the quadrilateral?\n\nPlea... | 2(d_1 + d_2) - P | polaris53k | 37,499 | 5/8 | null | mixed1to3 | 3 | 0.75 | 5,816 | 0 | false | false | 4 | 4 | 2 | 1 | 1 | d_1 + d_2 + d_3 + d_4 = p + q - P | 1 | confirmed | false |
[
{
"role": "user",
"content": "The radius of the circumcircle of the acute-angled triangle \\(ABC\\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \\(A\\), \\(C\\), and the orthocenter of triangle \\(ABC\\). Find \\(AC\\).\n\nPlease reason step by ... | \sqrt{3} | polaris53k | 7,832 | 1/8 | null | mixed1to3 | 3 | 0.75 | 19,941 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "There are 5 houses, numbered 1 to 5 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`, `Peter`, `Eric`, `Arnold`... | tea | polaris53k | 42,047 | 0/8 | null | mixed1to3 | 2 | 0.5 | 19,404 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "## [ Coordinate method in space ] Vectors (other).\n\nFind the angle between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$.\n\nPlease reason step by step, and put your final answer with... | \arccos\left( \dfrac{2\sqrt{105}}{35} \right) | polaris53k | 36,796 | 5/8 | null | mixed1to3 | 3 | 0.75 | 8,486 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "If every one-dimensional subspace of a Banach space $X$ is one-complemented, determine whether every two-dimensional subspace of $X$ is isometrically isomorphic to a Hilbert space. Answer true or false.\n\nPlease reason step by step, and put your final answer within \\boxed{}."... | False | deepmath103k | 63,792 | 8.5 | Mathematics -> Other | mixed1to3 | 2 | 0.5 | 7,204 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 1,2,\\cdots,10 中随机选出三个不同的数,它们两两互素的概率为__________.原始的答案是\\frac{m}{n}的形式,其中m、n是互质的。请给出最终m + n的值\n\nRemember... | 27 | dapo17k | 12,446 | null | null | mixed1to3 | 3 | 0.75 | 12,922 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 157 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Given the circle \\( M: 2x^{2} + 2y^{2} - 8x - 8y - 1 = 0 \\), and the line \\( l: x + y - 9 = 0 \\), let \\( A \\) be a point on \\( l \\). Construct \\(\\triangle ABC\\) such that \\(\\angle BAC = \\frac{\\pi}{4}\\), with side \\( AB \\) passing through the center of the circ... | [3,6] | polaris53k | 29,767 | 0/8 | null | mixed1to3 | 3 | 0.75 | 21,799 | 1 | false | false | 4 | 4 | 4 | 2 | 2 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAmong all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, ... | 21 | dapo17k | 6,916 | null | null | mixed1to3 | 2 | 0.5 | 16,901 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 22 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Determine the Euler characteristic of the edge word $abc^{-1}b^{-1}da^{-1}d^{-1}c$. Recall that the Euler characteristic is given by $X = V - E + F$, where $V$, $E$, and $F$ represent the number of vertices, edges, and faces, respectively.\n\nPlease reason step by step, and put... | -2 | deepmath103k | 1,884 | 5.0 | Mathematics -> Geometry -> Differential Geometry -> Other | mixed1to3 | 3 | 0.75 | 12,459 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Evaluate the integral: $$\\int_{-\\infty}^\\infty \\frac{e^{-x} \\, dx}{1-e^{-2x}}$$\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 0 | deepmath103k | 8,992 | 8.0 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | mixed1to3 | 3 | 0.75 | 12,523 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | \text{The integral diverges} | 1 | confirmed | true |
[
{
"role": "user",
"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: `Alice`, `Arnold`, `Eric`, `Pete... | pop | polaris53k | 45,949 | 0/8 | null | mixed1to3 | 3 | 0.75 | 15,194 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Let $g(x)$ be a continuous function for all real values of $x$ that satisfies the following equation:\n\n$$\\int^x_1 g(t)dt =\\int^0_{x} t^3g(t)dt +\\frac{x^{18}}{9}+\\frac{x^8}{4}+b$$\n\nFind the value of $b$.\n\nPlease reason step by step, and put your final answer within \\b... | -\dfrac{13}{36} | deepmath103k | 67,007 | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | mixed1to3 | 1 | 0.25 | 26,108 | 0 | false | false | 4 | 4 | 0 | 0 | 0 | b = \frac{7}{90} + \frac{4\ln 2}{3} - \frac{2\pi}{3\sqrt{3}} | 1 | uncertain | false |
[
{
"role": "user",
"content": "Consider two numerical sequences $X_n$ and $Y_n$. The sequence $X_n$ consists of random integers from one to six for each $n$. Define $Y_1 = X_1$, and for $n \\geq 2$, $$Y_n = X_n + \\frac{1}{Y_{n-1}}.$$ Determine the probability $P$ that $$\\frac{1+\\sqrt{3}}{2} \\leq Y_n \\le... | \dfrac{1}{5} | deepmath103k | 74,258 | 6.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | zero | 0 | 0 | 16,033 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "41. The road from home to school takes Seryozha 30 minutes. Once on the way, he remembered that he had forgotten a pen at home. Seryozha knew that if he continued on to school at the same speed, he would arrive there 9 minutes before the bell, but if he returned home for the pe... | \dfrac{1}{3} | polaris53k | 48,314 | 3/8 | null | mixed1to3 | 3 | 0.75 | 4,790 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Given the line $l$: $x-2y+2=0$ passes through the left focus $F\\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | \frac{2\sqrt{5}}{5} | polaris53k | 28,589 | 4/8 | null | mixed1to3 | 2 | 0.5 | 18,659 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ and $y$ be real numbers such that\n\\[\\frac{\\sin x}{\\cos y} + \\frac{\\sin y}{\\cos x} = 1 \\qu... | 137 | dapo17k | 6,719 | null | null | mixed1to3 | 2 | 0.5 | 11,252 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | 5 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\) in which \\(\\angle BAC=20^\\circ\\), \\(\\angle CAD=60^\\circ\\), \\(\\angle ADB=50^\\circ\\), and \\(\\angle BDC=10^\\circ\\). Find \\(\\angle ACB\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 80 | polaris53k | 25,613 | 3/8 | null | mixed1to3 | 3 | 0.75 | 13,668 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4\\%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded fo... | \frac{24}{7} | polaris53k | 2,521 | 2/8 | null | mixed1to3 | 2 | 0.5 | 19,842 | 1 | false | false | 4 | 4 | 2 | 2 | 0 | \dfrac{25}{7} | 2 | confirmed | true |
[
{
"role": "user",
"content": "In triangle \\(KLM\\), the ratio of the circumradius to the inradius is 3. The incircle touches the sides of triangle \\(KLM\\) at points \\(A\\), \\(B\\), and \\(C\\). Find the ratio of the area of triangle \\(KLM\\) to the area of triangle \\(ABC\\).\n\nPlease reason step by ... | 6 | polaris53k | 18,448 | 6/8 | null | mixed1to3 | 3 | 0.75 | 10,351 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Find the number of permutations \\(a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}\\) of the six integers from 1 to 6 such that for all \\(i\\) from 1 to 5, \\(a_{i+1}\\) does not exceed \\(a_{i}\\) by 1.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 309 | polaris53k | 31,011 | 0/8 | null | mixed1to3 | 3 | 0.75 | 17,659 | 0 | false | false | 4 | 4 | 1 | 1 | 0 | 231 | 1 | uncertain | false |
[
{
"role": "user",
"content": "A sequence of \\( n \\) positive integers is considered full if for each \\( k > 1 \\), \\( k \\) occurs only if \\( k-1 \\) occurs before the last occurrence of \\( k \\). How many full sequences are there for each \\( n \\)?\n\nPlease reason step by step, and put your final a... | n! | polaris53k | 13,942 | 0/8 | null | mixed1to3 | 2 | 0.5 | 24,606 | 1 | false | false | 4 | 3 | 3 | 3 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the ro... | 2031 | polaris53k | 27,597 | 0/8 | null | mixed1to3 | 3 | 0.75 | 14,962 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | 2023 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Are all compact subsets of \\( \\mathbb{R}^{\\mathbb{R}} \\) separable? Consider \\( \\mathbb{R} \\) with its usual topology and \\( \\mathbb{R}^{\\mathbb{R}} \\) with the topology of pointwise convergence. Provide a justification for your answer.\n\nPlease reason step by step,... | No | deepmath103k | 20,530 | 9.0 | Mathematics -> Calculus -> Differential Calculus -> Other | mixed1to3 | 2 | 0.5 | 10,620 | 0 | false | false | 4 | 4 | 4 | 0 | 4 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "In a tournament where each pair of teams played against each other twice, there were 4 teams participating. A win awarded two points, a draw one point, and a loss zero points. The team that finished in last place earned 5 points. How many points did the team that finished in fi... | 7 | polaris53k | 2,296 | 1/8 | null | mixed1to3 | 1 | 0.25 | 17,653 | 1 | false | false | 4 | 4 | 3 | 3 | 0 | 9 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Find a function \\( \\mu(x,y) \\) such that the differential equation \\( \\mu(x,y)(3x^2y+2xy+y^3)dx + \\mu(x,y)(x^2+y^2)dy = 0 \\) is exact.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | e^{3x} | deepmath103k | 52,756 | 4.5 | Mathematics -> Differential Equations -> Ordinary Differential Equations (ODEs) | mixed1to3 | 3 | 0.75 | 18,373 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"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... | stir fry | polaris53k | 42,604 | 0/8 | null | mixed1to3 | 2 | 0.5 | 22,400 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n由 $1,2, \\cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 $60$ 的圆排列的个数为 $\\qquad$.\n\nRemember to put your answer ... | 21600 | dapo17k | 11,155 | null | null | mixed1to3 | 3 | 0.75 | 15,837 | 1 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinit... | 75 | polaris53k | 21,770 | 3/8 | null | mixed1to3 | 2 | 0.5 | 13,594 | 0 | false | false | 4 | 1 | 0 | 0 | 0 | c \ge 100 | 1 | uncertain | false |
[
{
"role": "user",
"content": "\nAn integer sequence \\(\\left\\{a_{i, j}\\right\\}(i, j \\in \\mathbf{N})\\), where,\n\\[ \n\\begin{array}{l} \na_{1, n} = n^{n} \\quad (n \\in \\mathbf{Z}_{+}), \\\\ \na_{i, j} = a_{i-1, j} + a_{i-1, j+1} \\quad (i, j \\geq 1).\n\\end{array}\n\\]\n\nWhat is the unit digit of... | 4 | polaris53k | 30,960 | 0/8 | null | mixed1to3 | 2 | 0.5 | 26,526 | 1 | false | false | 4 | 4 | 2 | 2 | 0 | 8 | 2 | confirmed | true |
[
{
"role": "user",
"content": "A square-based pyramid has all edges 2 units long. We attach a pyramid to one of the side faces of the pyramid, where the side edges of the new pyramid are of equal length. The total length of the edges of the resulting solid is 18 units. What is the volume of the solid?\n\nPle... | 2\sqrt{2} | polaris53k | 38,712 | 0/8 | null | mixed1to3 | 1 | 0.25 | 18,497 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | \dfrac{2(6\sqrt{2}+\sqrt{39})}{9} | 2 | confirmed | true |
[
{
"role": "user",
"content": "In a cylinder with a height of 10 and a base radius of 1, two spheres are attached to the upper and lower bases. The sphere on the lower base is tangent to the bottom and the side of the cylinder, and the sphere on the upper base is tangent to the top and the side of the cylind... | 4\pi | deepmath103k | 90,479 | 6.0 | Mathematics -> Geometry -> Solid Geometry -> 3D Shapes | mixed1to3 | 1 | 0.25 | 18,485 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Let $(b_1,b_2,b_3,\\ldots,b_{14})$ be a permutation of $(1,2,3,\\ldots,14)$ for which\n$b_1>b_2>b_3>b_4>b_5>b_6>b_7>b_8 \\mathrm{\\ and \\ } b_8<b_9<b_{10}<b_{11}<b_{12}<b_{13}<b_{14}.$\nFind the number of such permutations.\n\nPlease reason step by step, and put your final an... | 1716 | polaris53k | 15,215 | 5/8 | null | mixed1to3 | 3 | 0.75 | 8,510 | 0 | false | false | 4 | 4 | 3 | 3 | 0 | 3003 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any positive integer $n$, define a function $f$ by \\[f(n)=2n+1-2^{\\lfloor\\log_2n\\rfloor+1}.\\] Let... | 8008 | dapo17k | 5,157 | null | null | zero | 0 | 0 | 29,177 | 1 | false | false | 4 | 4 | 3 | 3 | 0 | 1 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Determine whether the following statement is true or false: If matrix $A$ is nonsingular and matrix $B$ is singular, with $B^-$ being a generalized inverse of $B$, then $(AB)^- = B^- A^{-1}$.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | True | deepmath103k | 82,345 | 7.0 | Mathematics -> Algebra -> Linear Algebra -> Matrices | mixed1to3 | 2 | 0.5 | 13,405 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Determine whether the statement \"P(A) = P(B) if and only if A and B are logically equivalent\" holds true for any logical statements A and B.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | False | deepmath103k | 77,747 | 4.5 | Mathematics -> Discrete Mathematics -> Logic | mixed1to3 | 2 | 0.5 | 4,736 | 0 | false | false | 4 | 4 | 4 | 0 | 4 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, \"If you buy one more, you can get a 10% discount, which is $36 cheaper than now.\" Xiao Hua said, \"Then I'll buy one more, thank you.\"... | 486 | polaris53k | 31,385 | 2/8 | null | mixed1to3 | 1 | 0.25 | 12,171 | 0 | false | false | 4 | 4 | 2 | 2 | 0 | 32.4 | 1 | confirmed | true |
[
{
"role": "user",
"content": "Square \\(ABCD\\) has a side length of 4. Point \\(E\\) is the midpoint of \\(AB\\), and \\(F\\) is a variable point on side \\(BC\\). Triangles \\(\\triangle ADE\\) and \\(\\triangle DCF\\) are folded along \\(DE\\) and \\(DF\\) respectively, so that points \\(A\\) and \\(C\\)... | \frac{4\sqrt{5}}{5} | polaris53k | 12,184 | 0/8 | null | mixed1to3 | 2 | 0.5 | 17,200 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1}, l_{2}$, 其中 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_... | 16 | dapo17k | 12,619 | null | null | mixed1to3 | 3 | 0.75 | 15,200 | 1 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "An equilateral hexagon with side length 1 has interior angles $90^{\\circ}, 120^{\\circ}, 150^{\\circ}, 90^{\\circ}, 120^{\\circ}, 150^{\\circ}$ in that order. Find its area.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | \frac{3+\sqrt{3}}{2} | polaris53k | 18,736 | 6/8 | null | mixed1to3 | 3 | 0.75 | 16,237 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Calculate the flux of the vector field \\(\\mathbf{a} = y^2 \\mathbf{j} + z \\mathbf{k}\\) through the surface defined by \\(z = x^2 + y^2\\), bounded by the plane \\(z=2\\). Assume the normal vector is outward with respect to the region enclosed by the paraboloid.\n\nPlease re... | -2\pi | deepmath103k | 87,130 | 5.5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | mixed1to3 | 2 | 0.5 | 16,037 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Given a strictly increasing sequence of natural numbers $(p_n)$ and a positive sequence $(\\epsilon_n)$ decreasing to $0$, let $(x_n)$ be a sequence in the unit sphere of $l_2$ with the property that for any $j \\in \\mathbb{N}$ and for any $1 \\leq k \\leq j$, \\( \\sum_{i>p_k... | Yes | deepmath103k | 16,491 | 8.5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | mixed1to3 | 2 | 0.5 | 14,176 | 0 | false | false | 4 | 4 | 4 | 0 | 4 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "In triangle $ABC$, point $E$ is on side $BC$, $\\angle BAE = 20^\\circ$ and $\\angle EAC = 40^\\circ$. Determine the measure of angle $BEA$, in degrees.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 80^\circ | polaris53k | 51,310 | 1/8 | null | mixed1to3 | 2 | 0.5 | 19,850 | 0 | false | false | 4 | 4 | 1 | 1 | 0 | 100 | 2 | uncertain | false |
[
{
"role": "user",
"content": "Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 72000 | polaris53k | 19,221 | 5/8 | null | mixed1to3 | 3 | 0.75 | 7,868 | 0 | false | false | 4 | 4 | 4 | 1 | 3 | 0 | confirmed | false | |
[
{
"role": "user",
"content": "Given a set of points within a time range from 0 to N, select 4 points such that the distance between each selected point is maximized. For example, if the range is 0-10 and the points are {1, 2, 4, 5, 7, 9, 10}, determine the 4 points that maximize the distance between them.\n... | \{1, 4, 7, 10\} | deepmath103k | 19,856 | 3.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations | mixed1to3 | 3 | 0.75 | 8,014 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"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`, `Pete... | green | polaris53k | 34,640 | 2/8 | null | mixed1to3 | 3 | 0.75 | 16,210 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "There are 5 houses, numbered 1 to 5 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`, `Peter`, `Alice`, `Eric`,... | 3 | polaris53k | 42,404 | 0/8 | null | mixed1to3 | 3 | 0.75 | 13,390 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn $2020 \\times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetromino... | 2020 | dapo17k | 3,195 | null | null | zero | 0 | 0 | 15,034 | 0 | false | false | 4 | 3 | 3 | 3 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Given column vectors $v_1, v_2, v_3 \\in \\mathbb{R}^2$, if for all distinct $i, j \\in \\{1, 2, 3\\}$, the minimum eigenvalue of the matrix $v_iv_i^T + v_jv_j^T$ satisfies $\\lambda_{\\min}(v_iv_i^T+v_jv_j^T) \\leq 1$, determine an upper bound for $\\lambda_{\\min}(v_1v_1^T+v_... | 3 | deepmath103k | 55,648 | 6.5 | Mathematics -> Algebra -> Linear Algebra -> Matrices | zero | 0 | 0 | 16,019 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true | |
[
{
"role": "user",
"content": "Arrange the 5 numbers \\(2, 0, 1, 9, 2019\\) in any order to form an 8-digit number (the first digit cannot be 0). Determine the number of different 8-digit numbers that can be produced.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | 95 | polaris53k | 20,020 | 0/8 | null | zero | 0 | 0 | 8,352 | 0 | false | false | 4 | 4 | 4 | 4 | 0 | 0 | confirmed | true |
math_rlvr_filtered
Public math RL datasets (three in full, a fourth as one filtered file), each problem annotated with how often Qwen3-4B solves it in 4 attempts, plus a multiple-choice flag. The point is RL data selection: problems the model solves 1-3 times out of 4 carry GRPO-style learning signal; 4/4 problems are too easy for this model and 0/4 problems are either too hard or have unreliable reference answers.
| config | source | problems (all) |
train_nomc (multiple choice removed) |
pass@4 | 1-3 of 4 correct |
|---|---|---|---|---|---|
deepmath103k |
zwhe99/DeepMath-103K | 103,022 | 100,040 | 94.7% | 14,397 (14.4%) |
dapo17k |
zhuzilin/dapo-math-17k (deduplicated DAPO-Math-17k) | 17,398 | 17,379 | 89.7% | 3,167 (18.2%) |
polaris53k |
POLARIS-Project/Polaris-Dataset-53K | 53,291 | 52,206 | 84.0% | 14,549 (27.9%) |
Statistics in the last two columns are for train_nomc.
A fourth source is included as one filtered file, intellect2_pass0to3:
PrimeIntellect/INTELLECT-2-only-math (259,465 rows: NuminaMath-1.5
217,088 + DeepScaleR 42,377). It keeps only the problems Qwen3-4B solved 0-3 times out of 4, with multiple choice removed: 97,113 rows.
| n_correct (of 4) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| rows | 73,334 | 6,861 | 6,516 | 10,402 |
ds = load_dataset("clyu/math_rlvr_filtered", "intellect2_pass0to3", split="train")
signal = ds.filter(lambda r: r["n_correct"] >= 1) # the 23,779 problems with 1-3 of 4 correct
Over the whole labeled set (247,056 problems after multiple choice is removed), pass@4 is 70.3% and 9.6% of problems have 1-3 of 4 correct. Only problems Qwen3-4B solved 4/4 (149,943) and multiple choice (607) are left out of this file.
Two more configs build an RL training set from these labels, after a second model, GLM-5.3-Flash, re-checked the reference answers:
| config | rows | what it is |
|---|---|---|
rl_final (default) |
35,660 | the recommended RL training set: DeepMath 15,305 / Polaris 16,318 / DAPO 4,037 |
glm_labels |
45,988 | every candidate problem (Qwen3-4B 1-3/4 and 0/4) with its GLM-5.3-Flash validation columns |
from datasets import load_dataset
train = load_dataset("clyu/math_rlvr_filtered", "rl_final", split="train") # chat-format `prompt` + `label`
RL training set (rl_final) and GLM-5.3-Flash validation (glm_labels)
Candidate pool. Built from the three train_nomc splits:
- 1-3/4 pool (30,964 problems): Qwen3-4B solved the problem 1, 2 or 3 times out of 4.
- Polaris rows flagged by a corrected
answer_suspectheuristic were dropped (4). - Exact-duplicate problems across or within sources were merged, keeping one copy with priority DAPO > DeepMath > Polaris.
- Duplicate groups with conflicting answers were dropped (19 rows).
- Polaris rows flagged by a corrected
- 0/4 pool (15,024 problems): Qwen3-4B solved none of 4.
- Suspect answers and conflicting duplicates are kept but flagged.
GLM validation. zai-org/GLM-5.3-Flash (FP8, served with sglang 0.5.20, TP8) answered every candidate problem 4 times, with
reasoning_effort high, temperature 1.0, top_p 0.95, a 65,536-token cap and the same prompt as the Qwen labeling.
- Each answer was graded with the same rules as above (
glm_n_rule_correct). - If the rules marked an answer wrong, GLM itself (effort low, temperature 0) judged whether it and the reference answer are the SAME final
answer (
glm_n_judged_same).
glm_n_correct is the sum of the two, and the per-problem verdict is:
confirmed:glm_n_correct >= 2disputed:glm_n_correct == 0and at least 3 of the 4 answers agree on one alternative (glm_top_alt). This likely means a wrong reference answer, though GLM can also be consistently wrong.uncertain: everything else (one match, missing answers or answers that disagree with each other).
| pool | source | n | confirmed | uncertain | disputed |
|---|---|---|---|---|---|
| 1-3/4 | DeepMath | 14,221 | 85.3% | 11.9% | 2.8% |
| 1-3/4 | Polaris | 13,597 | 86.6% | 10.3% | 3.2% |
| 1-3/4 | DAPO | 3,146 | 87.5% | 9.6% | 2.9% |
| 0/4 | DeepMath | 5,210 | 79.9% | 16.1% | 4.0% |
| 0/4 | Polaris | 8,043 | 61.6% | 23.0% | 15.5% |
| 0/4 | DAPO | 1,771 | 56.2% | 22.4% | 21.4% |
In the 1-3/4 pool, the disputed share falls from about 5% (Qwen 1/4) to 2% (Qwen 3/4).
Rule-match requirement for 0/4. Many 0/4 confirmations come only from the SAME judge, not the rules: 4,497 problems, half of DeepMath's 0/4. In a spot check the judge was usually right, but these references are often things a rule-based reward cannot check:
- corrupted Polaris strings;
- statements to prove or inequalities;
- Yes/No answers where GLM adds prose.
So rl_final keeps a 0/4 problem only if GLM's answer rule-matches the label at least 2 of 4 times (glm_rule_confirmed).
rl_final selection.
- 1-3/4: every problem except GLM-
disputed(912 dropped;uncertainkept, since Qwen already matched these labels at least once).- Kept: 30,052 problems. By Qwen n_correct: 1/4 6,891, 2/4 8,172, 3/4 14,989.
- 0/4: only
glm_rule_confirmedand notdup_conflict. These are hard problems for Qwen3-4B whose labels a rule grader can match.- Kept: 5,608 problems. By source: DeepMath 1,477, Polaris 3,150, DAPO 981.
Rows are shuffled (seed 0). Qwen3-4B's mean accuracy on rl_final is 0.478, with 15.0k mean completion tokens.
Columns (rl_final, glm_labels)
| column | meaning |
|---|---|
prompt |
chat messages ([{"role": "user", "content": ...}]), exactly the prompt used for labeling |
label |
reference answer (string) |
source, source_idx |
source config (deepmath103k / dapo17k / polaris53k) and its idx |
difficulty, topic |
source fields as strings (see below); null when absent |
qwen_pool |
mixed1to3 or zero |
qwen_n_correct, qwen_mean_acc, qwen_mean_tokens, qwen_n_length_capped |
Qwen3-4B pass@4 labels (see below) |
answer_suspect |
corrected heuristic for corrupted reference answers (0/4 pool; always false in the 1-3/4 pool, where flagged rows were dropped) |
dup_conflict |
the problem has an exact duplicate with a different reference answer (0/4 pool only) |
glm_n_samples, glm_n_answered |
GLM samples (4) and how many produced a final answer |
glm_n_correct, glm_n_rule_correct, glm_n_judged_same |
GLM samples matching the label: total = rule-matched + judged SAME |
glm_top_alt, glm_top_alt_count |
GLM's most common non-matching answer and how many samples gave it |
glm_verdict |
confirmed / uncertain / disputed |
glm_rule_confirmed |
glm_n_rule_correct >= 2 |
from datasets import load_dataset
ds = load_dataset("clyu/math_rlvr_filtered", "polaris53k", split="train_nomc")
signal = ds.filter(lambda r: 1 <= r["n_correct"] <= 3)
How the labels were made
- Model:
Qwen/Qwen3-4B, thinking mode on, the model card's sampling (temperature 0.6, top_p 0.95, top_k 20), up to 32,768 new tokens (40,960-token context). Served with sglang; each of the 4 samples per problem used a distinct seed. - Prompt:
- DeepMath and Polaris: the problem text followed by
Please reason step by step, and put your final answer within \boxed{}. - DAPO: the dataset's own chat prompt, unchanged (it asks for
Answer: \boxed{...}on the last line).
- DeepMath and Polaris: the problem text followed by
- Grading:
math_verifyagainst the reference answer, with a fallback that compares the last\boxed{}(orAnswer:line) to the reference after light LaTeX normalization (\dfrac→\frac,\varnothing→\emptyset,\text{}, spacing, thousands separators, Yes/No case). For Polaris, an extra rule accepts the option letter for references like\textbf{(D)} 20\sqrt{2}.
Columns (per-source configs)
| column | meaning |
|---|---|
idx |
row index in the source file (DeepMath: concatenated parquet shards in order; DAPO: dapo-math-17k.jsonl; Polaris: polaris-data-53K.jsonl) |
question |
problem text (for DAPO: the user message, including DAPO's answer-format instruction) |
final_answer |
reference answer from the source dataset |
difficulty |
source field: DeepMath's 1-10 rating; Polaris's DeepSeek-R1-Distill-Qwen-7B pass rate (k/8); null for DAPO |
topic |
DeepMath's topic; null otherwise |
n_samples |
always 4 |
n_correct |
samples graded correct (0-4) |
pass_at_4 |
n_correct > 0 |
mean_acc |
n_correct / 4 |
n_length_capped |
samples that hit the 32k-token cap |
mean_tokens |
mean completion tokens over the 4 samples |
is_multiple_choice, mc_type |
multiple-choice flag and kind: options_listed (the question lists options A/B/C), tf_letter_label (a true/false question whose reference answer is A/B), letter_label (any other bare-letter reference answer) |
answer_suspect |
Polaris only: heuristic flag for corrupted reference answers (see caveats) |
train_nomc contains exactly the rows with is_multiple_choice == false. Removed: DeepMath 2,982 (2.89%), DAPO 19 (0.11%), Polaris 1,085 (2.04%).
Caveats
- 0/4 is the least reliable bucket. An audit of the problems where 3-4 samples agree on one answer but are graded wrong found grader or
reference-answer issues in roughly 1% of DeepMath problems and 1.5% or more of Polaris problems; DAPO, whose answers are all integers, is
essentially clean (about 0.05%). Typical causes:
YesvsTrue, function notation (f(x) = xvsf(n) = n), case in text answers, and corrupted Polaris references. They sit almost entirely in the 0/4 bucket; the 1-3/4 bucket is barely affected. - DeepMath true/false with letter answers. 664 DeepMath problems ask "true or false" but give
A/Bas the reference answer without saying which letter means what. All 664 are 0/4 here. The model's answers agree with A = True, B = False on 88.5% of them. They are taggedtf_letter_labeland excluded fromtrain_nomc. - Polaris reference answers are sometimes corrupted, apparently with LaTeX commands stripped (
\2,^2=\frac{1}{3}(...,\frac{}{}).answer_suspectflags some of these, but it is a heuristic with a known bug: it also flags valid half-open intervals such as(4,9]or[1,\infty)(155 of its 643 flags inallare such intervals), and it misses corruptions that look valid, such as2-1. - Multiple-choice detection is heuristic.
letter_labelalso catches a few problems whose answer is genuinely a letter-named object (for example "which player wins? B"); these are excluded too. - Labels reflect Qwen3-4B under the settings above. A different model, prompt or token budget will shift the difficulty buckets.
INTELLECT-2 (intellect2_pass0to3)
Labeling. Same model, sampling, token budget and grader as the other sources.
- The prompt is the dataset's
promptfollowed by the DeepMath-style\boxed{}instruction. - The reference answer is
verification_info.ground_truth. - 11,802 rows have no ground truth (
null) and were not labeled. They are not in this file.
Columns. As in the per-source configs, plus problem_id.
idxis the row index in the source's 7 parquet shards, concatenated in order.question,final_answer,problem_id.difficultyis the source'ssolve_rate_qwen_r1_distill_7b, as a float.n_samples,n_correct,pass_at_4,mean_acc,n_length_capped,mean_tokens.answer_suspectuses the corrected heuristic, which no longer flags valid half-open intervals.
Rows: NuminaMath-1.5 84,822 / DeepScaleR 12,291.
Caveats.
- The 0/4 bucket is large and noisy: 73,334 rows, 76% of the file. Its share of all labeled problems is 29.7%, against 6.1% for DeepMath, 9.9% for DAPO and 15.9% for Polaris.
- Most 0/4 rows are consistent disagreements. In an audit of a quarter of the rollouts, 72% of 0/4 problems had at least 3 of 4 samples agreeing on one answer that differs from the reference.
- Grader false negatives are rare (0.35% of samples).
- About 5% of those consensus disagreements are visibly corrupted references, with LaTeX commands stripped (
\frac{^2+n^2}{n-},(,b,)\in\{\dots\}). - Most of the rest are genuine disagreements: either a wrong reference or a problem the model consistently gets wrong. The labels alone cannot tell which.
answer_suspectflags 2,257 rows, almost all NuminaMath-1.5 and almost all 0/4. It is a lower bound on corrupted references.- Overlap with the other sources. About 28.7k INTELLECT-2 problems, before filtering, are exact (whitespace-normalized) duplicates of DeepMath-103K or Polaris-53K problems. Most are DeepScaleR rows that also appear in Polaris. Deduplicate before mixing configs.
- The 1-3/4 rows (23,779) are the reliable RL signal. 0/4 rows have not been checked with GLM-5.3-Flash, unlike the
glm_labelspool.
Licenses
Each config inherits the license of its source: DeepMath-103K is MIT; DAPO-Math-17k and Polaris-Dataset-53K are Apache-2.0. The added
annotation columns are released under the same terms as the corresponding source. In rl_final and glm_labels, each row follows the
license of its source; glm_top_alt is GLM-5.3-Flash output (MIT-licensed model).
INTELLECT-2-only-math does not declare a license on its dataset card. Its rows come from NuminaMath-1.5 (Apache-2.0) and DeepScaleR
(MIT). Check the upstream terms before redistributing intellect2_pass0to3.
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