Datasets:

MatouK98
/

math_rlvr_filtered_audited

prompt listlengths 1 1	label stringlengths 1 136	source stringclasses 3 values	source_idx int64 0 103k	difficulty stringclasses 26 values	topic stringclasses 77 values	qwen_pool stringclasses 2 values	qwen_n_correct int64 0 2	qwen_mean_acc float64 0 0.5	qwen_mean_tokens int64 774 32.8k	qwen_n_length_capped int64 0 4	answer_suspect bool 1 class	dup_conflict bool 1 class	glm_n_samples int64 4 4	glm_n_answered int64 0 4	glm_n_correct int64 0 4	glm_n_rule_correct int64 0 4	glm_n_judged_same int64 0 4	glm_top_alt stringlengths 0 802	glm_top_alt_count int64 0 3	glm_verdict stringclasses 2 values	glm_rule_confirmed bool 2 classes	hf_row_1based int64 2 35.7k	audit_state stringclasses 3 values
[ { "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	2	previously_retained
[ { "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	7	previously_retained
[ { "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	9	previously_retained
[ { "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	11	previously_retained
[ { "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	15	previously_retained
[ { "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	18	previously_retained
[ { "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	21	previously_retained
[ { "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	22	previously_retained
[ { "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 $Answer is the answer to the problem.\n\nA rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of th...	2007	dapo17k	5,338	null	null	mixed1to3	1	0.25	17,477	0	false	false	4	4	4	4	0		0	confirmed	true	23	previously_retained
[ { "role": "user", "content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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	26	previously_retained
[ { "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	29	previously_retained
[ { "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	33	previously_retained
[ { "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	35	previously_retained
[ { "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	37	previously_retained
[ { "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	39	previously_retained
[ { "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	41	previously_retained
[ { "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	42	previously_retained
[ { "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	44	previously_retained
[ { "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	49	previously_retained
[ { "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	51	previously_retained
[ { "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	54	previously_retained
[ { "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	56	previously_retained
[ { "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	59	previously_retained
[ { "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	64	previously_retained
[ { "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	66	previously_retained
[ { "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	67	previously_retained
[ { "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	69	previously_retained
[ { "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	72	previously_retained
[ { "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	74	previously_retained
[ { "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	75	previously_retained
[ { "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	77	previously_retained
[ { "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	80	previously_retained
[ { "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	81	previously_retained
[ { "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	82	previously_retained
[ { "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	84	previously_retained
[ { "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	85	previously_retained
[ { "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	86	previously_retained
[ { "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	87	previously_retained
[ { "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	88	previously_retained
[ { "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	91	previously_retained
[ { "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	92	previously_retained
[ { "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	98	previously_retained
[ { "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 $\\lambda_{\\min}(v_1v_1^T+v_...	3	deepmath103k	55,648	6.5	Mathematics -> Algebra -> Linear Algebra -> Matrices	zero	0	0	16,019	0	false	false	4	4	4	4	0		0	confirmed	true	99	previously_retained
[ { "role": "user", "content": "Arrange the 5 numbers \$2, 0, 1, 9, 2019\$ in any order to form an 8-digit number (the first digit cannot be 0). Determine the number of different 8-digit numbers that can be produced.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	95	polaris53k	20,020	0/8	null	zero	0	0	8,352	0	false	false	4	4	4	4	0		0	confirmed	true	100	previously_retained
[ { "role": "user", "content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe twenty-\ffirst century began on January $1$, $2001$ and runs through December $31$, $2100$. Note that ...	44	dapo17k	8,035	null	null	mixed1to3	2	0.5	19,186	1	false	false	4	4	0	0	0	43	2	uncertain	false	105	previously_retained
[ { "role": "user", "content": "Is it possible to have a ring $A$, with identity, such that $A^m \\cong A^n$ as left $A$-modules with $m \\neq n$? Provide a proof or counterexample.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	Yes	deepmath103k	24,367	8.0	Mathematics -> Algebra -> Abstract Algebra -> Ring Theory	mixed1to3	2	0.5	8,118	0	false	false	4	4	4	0	4		0	confirmed	false	106	previously_retained
[ { "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: `Eric`, `Alice`, `Peter`, `Bob`,...	hamster	polaris53k	53,219	0/8	null	mixed1to3	2	0.5	14,954	0	false	false	4	4	3	3	0	\text{cannot be determined}	1	confirmed	true	107	previously_retained
[ { "role": "user", "content": "A fair die is tossed repeatedly until the last six outcomes form the pattern 131131. What is the expected number of rolls needed to achieve this pattern?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	46878	deepmath103k	28,165	6.0	Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations	mixed1to3	2	0.5	18,126	0	false	false	4	4	4	4	0		0	confirmed	true	110	previously_retained
[ { "role": "user", "content": "In a regular quadrilateral pyramid $P-ABCD$, if $\\angle APC = 60^{\\circ}$, find the cosine of the dihedral angle between planes $A-PB-C$.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	$-\frac{1}{7}$	polaris53k	27,104	0/8	null	mixed1to3	2	0.5	10,985	0	false	false	4	4	3	3	0	\dfrac{1}{7}	1	confirmed	true	112	previously_retained
[ { "role": "user", "content": "Ostrovsky $M$.\n\nA number from 1 to 144 is guessed. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. You have to pay 2 rubles for a \"yes\" answer and 1 ruble for a \"no\" answer. What is the minimum amo...	11	polaris53k	35,754	3/8	null	mixed1to3	2	0.5	15,098	0	false	false	4	4	4	4	0		0	confirmed	true	113	previously_retained
[ { "role": "user", "content": "What is the largest number of starting clues in a Sudoku puzzle that does not guarantee a unique solution?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	77	deepmath103k	50,488	7.5	Mathematics -> Discrete Mathematics -> Combinatorics	mixed1to3	2	0.5	5,464	0	false	false	4	4	4	4	0		0	confirmed	true	115	previously_retained
[ { "role": "user", "content": "For an integer \$ n \\geq 3 \$, we say that \$ A = \\left( a_1, a_2, \\ldots, a_n \\right) \$ is an \$ n \$-list if every \$ a_k \$ is an integer in the range \$ 1 \\leq a_k \\leq n \$. For each \$ k = 1, \\ldots, n-1 \$, let \$ M_k \$ be the minimal possible non-...	4(n-1)	polaris53k	4,203	1/8	null	mixed1to3	2	0.5	21,116	0	false	false	4	4	4	4	0		0	confirmed	true	121	previously_retained
[ { "role": "user", "content": "16. Let $O$ be the centre of the circumcircle of $\\triangle A B C, P$ and $Q$ the midpoints of $A O$ and $B C$, respectively. Suppose $\\angle C B A=4 \\angle O P Q$ and $\\angle A C B=6 \\angle O P Q$. Find the size of $\\angle O P Q$ in degrees.\n\nPlease reason step by step...	12	polaris53k	44,069	2/8	null	mixed1to3	1	0.25	26,212	2	false	false	4	4	4	4	0		0	confirmed	true	122	previously_retained
[ { "role": "user", "content": "Find the sum of the smallest and largest possible values for \$ x \$ which satisfy the following equation.\n\\[ 9^{x+1} + 2187 = 3^{6x - x^2} \\]\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	5	polaris53k	7,957	1/8	null	mixed1to3	2	0.5	25,720	0	false	false	4	4	4	4	0		0	confirmed	true	123	previously_retained
[ { "role": "user", "content": "Determine whether the following inequality holds for all continuous functions $f$ and $g$ on $[0,1]$:\n\\[ \\sup_{x \\in [0,1]} \\left\\{ \\int_{0}^{x} \|f(t) - g(t)\| \\, dt \\right\\} \\leq \\sup_{x \\in [0,1]} \\left\\{ \|f(x) - g(x)\| \\right\\}. \\]\n\nPlease reason step by st...	Yes	deepmath103k	41,709	7.5	Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals	mixed1to3	2	0.5	7,118	0	false	false	4	4	4	0	4		0	confirmed	false	124	previously_retained
[ { "role": "user", "content": "Determine the values of \$ \\beta \$ for which the series \$ \\sum_{n=1}^{\\infty} \\left( \\sqrt{1+\\frac{(-1)^n}{n^\\beta}} - 1 \\right) \$ converges.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	\beta > \dfrac{1}{2}	deepmath103k	12,659	6.5	Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions	mixed1to3	2	0.5	11,390	0	false	false	4	4	4	1	3		0	confirmed	false	125	previously_retained
[ { "role": "user", "content": "Let \$X_t\$ be a Brownian motion and \$a < 0\$. Define the stopping time \$T_x = \\inf\\{t \\geq 0: X_t + at \\geq x\\}\$. If \$Y = \\sup_{t \\geq 0}(X_t + at) = x\$, determine whether \$T_x < \\infty\$.\n\nPlease reason step by step, and put your final answer within ...	T_x < \infty	deepmath103k	43,451	8.0	Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other	zero	0	0	9,412	0	false	false	4	4	4	3	1		0	confirmed	true	126	previously_retained
[ { "role": "user", "content": "Consider the sequence of integers \$ a_{1}, a_{2}, a_{3}, \\ldots \$ defined by \$ a_{1} = 1 \$ and for \$ n \\geq 1 \$, \$ a_{n+1} \$ is the smallest integer greater than \$ a_{n} \$ such that \$ a_{i} + a_{j} \\neq 3a_{k} \$ for any \$ i, j, \$ and \$ k \$ in ...	4494	deepmath103k	85,038	6.5	Mathematics -> Number Theory -> Other	mixed1to3	1	0.25	28,807	2	false	false	4	4	4	4	0		0	confirmed	true	129	previously_retained
[ { "role": "user", "content": "Find the smallest prime number \$ p \$ such that 7 is the least quadratic nonresidue modulo \$ p \$.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	71	deepmath103k	7,825	5.5	Mathematics -> Number Theory -> Prime Numbers	mixed1to3	1	0.25	16,223	0	false	false	4	4	3	3	0	241	1	confirmed	true	130	previously_retained
[ { "role": "user", "content": "Let $(x_1,...,x_n)$ and $(y_1,...,y_n)$ be two different tuples of positive real numbers such that $x_1 \\times \\dots \\times x_n = y_1 \\times \\dots \\times y_n = c$. Determine if the following inequality is true:\n\\[ \\left(\\frac{x_1+y_1}{2}\\right)\\times\\cdots\\times \...	True	deepmath103k	24,767	6.0	Mathematics -> Other	mixed1to3	1	0.25	6,019	0	false	false	4	4	4	0	4		0	confirmed	false	132	previously_retained
[ { "role": "user", "content": "## - Problem 3\n\nLet $A B C D$ be a convex quadrilateral. Let $P$ be the intersection of $A C$ and $B D$. The angle $\\angle A P D=60^{\\circ}$. Let $E, F, G$ and $H$ be the midpoints of the sides $A B, B C, C D$ and $D A$ respectively. Find the largest positive real number $k...	\sqrt{3} + 1	polaris53k	38,527	0/8	null	zero	0	0	18,953	0	false	false	4	4	4	3	1		0	confirmed	true	133	previously_retained
[ { "role": "user", "content": "\nA finite sequence of numbers \$ x_{1}, x_{2}, \\ldots, x_{N} \$ possesses the following property:\n\n\\[ x_{n+2} = x_{n} - \\frac{1}{x_{n+1}} \\text{ for all } 1 \\leq n \\leq N-2. \\]\n\nFind the maximum possible number of terms in this sequence if \$ x_{1} = 20 \$ and \...	322	polaris53k	4,251	2/8	null	mixed1to3	2	0.5	22,958	2	false	false	4	4	4	4	0		0	confirmed	true	135	previously_retained
[ { "role": "user", "content": "27 identical dice were glued together to form a $3 \\times 3 \\times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?\n\nPlease reason step by step, and put your ...	189	polaris53k	21,848	2/8	null	mixed1to3	2	0.5	20,774	0	false	false	4	4	4	4	0		0	confirmed	true	136	previously_retained
[ { "role": "user", "content": "Find the number of subgroups in the dihedral group $D_{16}$ that contain the stabilizer of a vertex in the action of $D_{16}$ on the vertices of an octagon.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	4	deepmath103k	78,286	6.0	Mathematics -> Algebra -> Abstract Algebra -> Group Theory	zero	0	0	17,782	0	false	false	4	4	3	3	0	5	1	confirmed	true	137	previously_retained
[ { "role": "user", "content": "Consider a hemispherical vase closed with a flat lid. Inside the vase, there are four identical oranges touching the vase, and one grapefruit touching all four oranges. Determine if it is true that all four points of contact of the grapefruit with the oranges necessarily lie in...	True	deepmath103k	88,918	7.0	Mathematics -> Geometry -> Solid Geometry -> 3D Shapes	mixed1to3	1	0.25	7,950	0	false	false	4	4	4	0	4		0	confirmed	false	141	previously_retained
[ { "role": "user", "content": "Let \$\\mathcal{P}_{1}, \\mathcal{P}_{2}, \\mathcal{P}_{3}\$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the \$\\mathcal{P}_{i}\$. In other words, find the maximum number of points that can lie on two...	12	polaris53k	14,691	4/8	null	mixed1to3	2	0.5	14,266	0	false	false	4	2	2	2	0		0	confirmed	true	142	previously_retained
[ { "role": "user", "content": "In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	64	polaris53k	25,395	0/8	null	zero	0	0	22,946	0	false	false	4	2	2	2	0		0	confirmed	true	143	previously_retained
[ { "role": "user", "content": "Let \$ N \$ be the number of triples of positive integers \$ (a, b, c) \$ satisfying\n\\[ a \\leq b \\leq c, \\quad \\operatorname{gcd}(a, b, c)=1, \\quad a b c=6^{2020} . \\]\n\nCompute the remainder when \$ N \$ is divided by 1000.\n\nPlease reason step by step, and put...	602	polaris53k	23,968	0/8	null	mixed1to3	1	0.25	27,730	0	false	false	4	4	3	3	0	2	1	confirmed	true	144	previously_retained
[ { "role": "user", "content": "Solve the equation \$x = 100e^{-x/100}\$ for \$x\$.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	56.714	deepmath103k	67,430	5.0	Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions	mixed1to3	2	0.5	9,402	0	false	false	4	4	4	0	4		0	confirmed	false	150	previously_retained
[ { "role": "user", "content": "Provide an example of an invertible linear transformation $T: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ such that there is no linear transformation $S: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ with $e^S = T$. What is the transformation $T$?\n\nPlease reason step by step, and p...	\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}	deepmath103k	37,892	8.0	Mathematics -> Algebra -> Linear Algebra -> Linear Transformations	mixed1to3	1	0.25	5,018	0	false	false	4	4	1	0	1	T = -I: \mathbb{R}^3 \to \mathbb{R}^3, \qquad T(v) = -v	1	uncertain	false	154	previously_retained
[ { "role": "user", "content": "Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.\nA) $\\frac{70}{7}$\nB) $\\frac{75}{7}$\nC) $\\frac{80}{7}$\nD) $10$\nE) $15$\n\nPlease reason step by step, and put your final answer within \\boxed{}." ...	\frac{75}{7}	polaris53k	31,074	0/8	null	zero	0	0	7,356	0	false	false	4	4	4	3	1		0	confirmed	true	155	previously_retained
[ { "role": "user", "content": "444. Eight hockey teams play against each other in a round-robin tournament to determine the final four. What is the minimum number of points that guarantees a team's advancement to the final four?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ...	11	polaris53k	35,407	1/8	null	mixed1to3	1	0.25	19,490	0	false	false	4	4	4	4	0		0	confirmed	true	158	previously_retained
[ { "role": "user", "content": "In a plane, 36 points are marked, with no three points being collinear. Some pairs of these points are connected by segments such that no more than 3 segments originate from any point. Determine the maximum number of distinct closed 4-segment broken lines that can be formed, wh...	54	deepmath103k	92,678	6.0	Mathematics -> Discrete Mathematics -> Combinatorics	mixed1to3	2	0.5	23,260	0	false	false	4	3	3	3	0		0	confirmed	true	160	previously_retained
[ { "role": "user", "content": "Let \$p\$ be a prime number and \$\\mathbb{F}_{p}\$ be the field of residues modulo \$p\$. Let \$W\$ be the smallest set of polynomials with coefficients in \$\\mathbb{F}_{p}\$ such that the polynomials \$x+1\$ and \$x^{p-2}+x^{p-3}+\\cdots+x^{2}+2x+1\$ are in \\(...	p!	polaris53k	9,153	1/8	null	mixed1to3	1	0.25	22,208	0	false	false	4	4	4	4	0		0	confirmed	true	162	previously_retained
[ { "role": "user", "content": "2. The chord $AB$ divides a circle with radius $r$ into two arcs in the ratio $1:2$. In the larger segment, a square is inscribed such that one of its sides lies on this chord. Express the length of the side of the square in terms of the radius $r$.\n\nPlease reason step by ste...	\dfrac{(2 + \sqrt{19})r}{5}	polaris53k	44,729	1/8	null	mixed1to3	2	0.5	10,167	0	false	false	4	4	2	2	0	s = \dfrac{(\sqrt{19}-2)\,r}{5}	1	confirmed	true	163	previously_retained
[ { "role": "user", "content": "Let $p$ be a prime number. What is the condition for the existence of an integer $k$ such that $k^{2} \\equiv -1 \\pmod{p}$?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	p = 2 \text{ or } p \equiv 1 \pmod{4}	polaris53k	49,470	4/8	null	mixed1to3	2	0.5	8,868	0	false	false	4	4	4	3	1		0	confirmed	true	164	previously_retained
[ { "role": "user", "content": "[ Transfer of side, diagonal, etc.] The Law of Sines\n\nIn trapezoid $ABCD (AB \\\| CD)$, $AB + CD = b$, diagonals $AC$ and $BD$ are related by $5AC = 3BD$. Find the area of the trapezoid if $\\angle BAC = 2 \\angle DBA$.\n\nPlease reason step by step, and put your final answer ...	\dfrac{5\sqrt{11}}{64} b^2	polaris53k	46,187	3/8	null	mixed1to3	2	0.5	23,254	2	false	false	4	4	4	4	0		0	confirmed	true	171	previously_retained
[ { "role": "user", "content": "Consider a solution \$\\phi_t(x)\$ of the differential equation \$x' = f(x)\$, where \$x \\in \\mathbb{R}^n\$, \$f: \\mathbb{R}^n \\to \\mathbb{R}^n\$, and \$f\$ is continuously differentiable (\$f \\in C^1\$). If the solution converges to a fixed point \$x_0\$ as...	Yes	deepmath103k	38,208	6.5	Mathematics -> Differential Equations -> Ordinary Differential Equations (ODEs)	mixed1to3	1	0.25	4,170	0	false	false	4	4	4	0	4		0	confirmed	false	173	previously_retained
[ { "role": "user", "content": "Let $\\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\\angle ABC$ and $\\angle ACB$ with $\\Gamma$. If $EF$ is tangent to the incircle $\\gamma$ of $\\triangle ABC$, then find the value of $\\angle BAC$.\n\nPlease r...	60^\circ	polaris53k	52,905	4/8	null	mixed1to3	1	0.25	24,580	0	false	false	4	4	4	4	0		0	confirmed	true	174	previously_retained
[ { "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`, `Arnold`, `Alice`, `Eric`...	yellow	polaris53k	42,344	0/8	null	mixed1to3	1	0.25	17,919	0	false	false	4	3	3	3	0		0	confirmed	true	176	previously_retained
[ { "role": "user", "content": "XI OM - I - Task 3\n\nEach side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles....	\dfrac{2}{9} S	polaris53k	45,195	1/8	null	mixed1to3	2	0.5	17,391	0	false	false	4	3	3	3	0		0	confirmed	true	178	previously_retained
[ { "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\nPoints $A,$ $B,$ $C,$ and $D$ are equally spaced along a line such that $AB = BC = CD.$ A point $P$ is lo...	43	dapo17k	6,662	null	null	zero	0	0	31,887	3	false	false	4	4	2	2	0	49	2	confirmed	true	179	previously_retained
[ { "role": "user", "content": "Let $\\Omega$ be a circle of radius 8 centered at point $O$, and let $M$ be a point on $\\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\\Omega$, or such that there exists some rectangle $A B C D$ containing $P$ whose center is on $\\Omega$ with $A...	164+64 \pi	polaris53k	31,186	0/8	null	mixed1to3	2	0.5	17,134	0	false	false	4	3	3	3	0		0	confirmed	true	180	previously_retained
[ { "role": "user", "content": "Let \$ P \$ be a point selected uniformly at random in the cube \$[0,1]^{3}\$. The plane parallel to \$ x+y+z=0 \$ passing through \$ P \$ intersects the cube in a two-dimensional region \$\\mathcal{R}\$. Let \$ t \$ be the expected value of the perimeter of \\(\\ma...	12108	polaris53k	24,614	2/8	null	mixed1to3	2	0.5	13,530	0	false	false	4	4	3	3	0	9809	1	confirmed	true	181	previously_retained
[ { "role": "user", "content": "Suppose $A:(\\mathbb{C}^n,+,\\diamond) \\to (\\mathbb{C}^n,+,\\cdot)$ is a unital homomorphism. Is $A$ necessarily invertible?\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	No	deepmath103k	8,770	8.0	Mathematics -> Algebra -> Abstract Algebra -> Field Theory	mixed1to3	2	0.5	12,702	0	false	false	4	4	4	0	4		0	confirmed	false	182	previously_retained
[ { "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\nDraw an equilateral triangle with center $O$. Rotate the equilateral triangle $30^\\circ, 60^\\circ, 90^\...	102	dapo17k	7,503	null	null	zero	0	0	19,057	0	false	false	4	4	4	4	0		0	confirmed	true	183	previously_retained
[ { "role": "user", "content": "Approximate the integral \$ \\int_3^4 \\frac{x}{\\sqrt{(7x-10-x^2)^3}}\\,dx \$.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	1.10	deepmath103k	52,868	6.0	Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable	mixed1to3	2	0.5	19,754	0	false	false	4	4	3	0	3	\dfrac{4\sqrt{2}}{9}	1	confirmed	false	184	previously_retained
[ { "role": "user", "content": "Two players take turns filling in the blanks of five equations of the form \$ x^{2} + \\ldots x + \\ldots = 0 \$ with natural numbers from 1 to 10, each used exactly once. The first player aims to maximize the number of equations with two distinct roots, while the second play...	3	deepmath103k	89,654	7.0	Mathematics -> Algebra -> Algebra -> Equations and Inequalities	mixed1to3	1	0.25	21,900	0	false	false	4	2	2	2	0		0	confirmed	true	188	previously_retained
[ { "role": "user", "content": "Does weak convergence of a sequence of functions in $L^2$ and uniform boundedness of the sequence in $L^\\infty$ norm imply the existence of a subsequence that converges pointwise almost everywhere?\n\nPlease reason step by step, and put your final answer within \\boxed{}." }...	No	deepmath103k	75,896	8.5	Mathematics -> Other	mixed1to3	1	0.25	12,358	0	false	false	4	4	3	0	3	\text{Yes — in fact the whole sequence converges pointwise a.e. to the weak limit } f.	1	confirmed	false	190	previously_retained
[ { "role": "user", "content": "Determine all real numbers $a$ for which there exists a function $f: \\mathbb{R} \\to \\mathbb{R}$ satisfying the equation \$ f(\\sin x) + a f(\\cos x) = \\cos 2x \$ for all real numbers $x$. \$ \\text{(I. Voronovich)} \$\n\nPlease reason step by step, and put your final an...	a \neq 1	deepmath103k	87,440	6.0	Mathematics -> Precalculus -> Functions	mixed1to3	2	0.5	14,790	0	false	false	4	4	3	3	0	a \in \{-1,\ 0\}	1	confirmed	true	197	previously_retained
[ { "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\nThere is a very popular race course where runners frequently go for a daily run. Assume that all runners r...	225	dapo17k	7,846	null	null	zero	0	0	15,385	0	false	false	4	3	3	3	0		0	confirmed	true	198	previously_retained
[ { "role": "user", "content": "There are 207 different cards with numbers $1, 2, 3, 2^{2}, 3^{2}, \\ldots, 2^{103}, 3^{103}$ (each card has one number, and each number appears exactly once). How many ways can you choose 3 cards such that the product of the numbers on the selected cards is a square of an inte...	267903	polaris53k	20,064	0/8	null	mixed1to3	1	0.25	19,364	0	false	false	4	4	4	4	0		0	confirmed	true	199	previously_retained
[ { "role": "user", "content": "Given the probability of failing on the first attempt and succeeding on the second attempt to guess the last digit of a phone number is requested, calculate the probability of this event.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	\frac{1}{10}	polaris53k	1,818	5/8	null	zero	0	0	6,640	0	false	false	4	4	4	4	0		0	confirmed	true	200	previously_retained
[ { "role": "user", "content": "Assume ZFC. Let $B \\subseteq \\mathbb{R}$ be a set that is not Borel-measurable. Can one conclude that $B$ necessarily has the cardinality of the continuum without assuming either the continuum hypothesis or its negation?\n\nPlease reason step by step, and put your final answe...	No	deepmath103k	10,470	9.0	Mathematics -> Other	mixed1to3	2	0.5	12,038	0	false	false	4	4	4	0	4		0	confirmed	false	201	previously_retained
[ { "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 farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections as shown in...	84	dapo17k	15,239	null	null	mixed1to3	2	0.5	22,090	0	false	false	4	4	3	3	0	66	1	confirmed	true	202	previously_retained
[ { "role": "user", "content": "Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.\nPlease give the answer directly without any intermediate steps.\n\nPlease reason step by step, and put y...	134	polaris53k	39,853	4/8	null	mixed1to3	2	0.5	25,264	1	false	false	4	4	2	2	0	336	1	confirmed	true	204	previously_retained
[ { "role": "user", "content": "Determine the Galois group of the polynomial \$ P(X) = (X^4 - 2)(X^2 + 2) \$ over \$ \\mathbb{Q} \$.\n\nPlease reason step by step, and put your final answer within \\boxed{}." } ]	D_4	deepmath103k	6,805	7.5	Mathematics -> Algebra -> Abstract Algebra -> Field Theory	mixed1to3	1	0.25	14,089	0	false	false	4	4	4	0	4		0	confirmed	false	206	previously_retained
[ { "role": "user", "content": "Thirty identical toothpicks were used to create the figure below. There are over 25 triangles in the figure. What is the fewest number of toothpicks that could be removed so that no triangles remain?\n\n[asy]\ndraw((0,0)--(8,0), linewidth(1));\ndraw(2dir(60)--(2dir(60)+(6,0)), ...	10	polaris53k	19,542	2/8	null	mixed1to3	1	0.25	15,445	0	false	false	4	3	4	4	0		0	confirmed	true	210	previously_retained
[ { "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已知四面体 $A B C D$ 的外接球 $\\Gamma$ 的半径为 3, 球心为 $O$, 其中 $A B$ 为球 $\\Gamma$ 的直径, $C D=3, \\angle A O C=\\theta \...	27	dapo17k	12,954	null	null	mixed1to3	2	0.5	19,283	0	false	false	4	4	4	4	0		0	confirmed	true	213	previously_retained
[ { "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`, `Eric`, `Carol`, `Bob`...	6	polaris53k	42,646	0/8	null	mixed1to3	2	0.5	17,622	0	false	false	4	4	4	4	0		0	confirmed	true	215	previously_retained

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Math RLVR: audited training and excluded records

This is an audited partition of the 35,660 records in clyu/math_rlvr_filtered, configuration rl_final, split train, pinned to revision 9106f86f6ab5159a31627af27233ef2841b27a94.

Splits

train: 18,106 records for math RLVR training, selected from the clean pool with qwen_n_correct in {0, 1, 2}.
train_full: 31,838 records, preserving the previous full clean training split exactly.
excluded: 3,822 excluded records, with a per-record Chinese reason. These include 3,669 remaining mathematical/question/proof exclusions and 153 uncertain records excluded by an explicit conservative selection policy. An uncertainty-policy exclusion does not assert that the reference is false.

train is a subset of train_full: the two overlap intentionally. Use train for the selected 18k training set; use train_full to recover the previous 31.8k clean pool. These are selection views, not independent train/test evaluation splits. The 13,732 rows with three Qwen-correct samples remain in train_full and are not quality exclusions.

Fields

All original fields are preserved, including prompt (role/content messages), label (the original reference answer), source, source_idx, and the existing Qwen/GLM diagnostic metadata. Question text and reference answers are unchanged.

All splits share the same columns. Additional fields:

hf_row_1based: original 1-based row number in the pinned source split.
exclusion_reason: Chinese explanation; an empty string in train and train_full.
exclusion_basis: comma-separated exclusion categories; empty in train and train_full. uncertainty_policy identifies the 153 conservatively removed uncertain records.
audit_state: provenance of the final selection decision.

For training, use prompt and label from the train split. Exclusion reasons and audit metadata are documentation, not model input or target text.

Selection history

The full initial audit retained 31,766 records and excluded 3,894. GPT-6.1 Sol with high reasoning effort then independently rechecked those 3,894 excluded records, with only the question/reference pair supplied. This was a recheck of exclusions, not a fresh GPT-6.1 review of every retained row. A focused answer-adequacy/proof check covered 64 restoration candidates. After adjudication, 225 records were restored. The final user policy then removed all 153 uncertain restorations, leaving 72 restorations: 25 judged usable and 47 retained under a reasonable standard interpretation. The original 31,766 retained records remain in train_full.

Retained records are not a mathematical guarantee of zero residual errors. The source dataset's attribution and applicable source terms continue to apply.

Loading

from datasets import load_dataset

data = load_dataset("MatouK98/math_rlvr_filtered_audited")
train = data["train"]           # 18,106
train_full = data["train_full"] # 31,838
excluded = data["excluded"]    # 3,822
print(excluded[0]["exclusion_reason"])

manifest.json records file hashes and the exact source revision.

Qwen3-4B difficulty selection

The default train split contains only rows with qwen_n_correct equal to 0, 1 or 2. The respective counts are 4,882, 5,924 and 7,300. Every question, reference answer, metadata field and original row order is preserved. Configuration qwen3_4b_correct_0_2 provides the same 18,106 rows as a named subset, also with a single train split.

from datasets import load_dataset

train = load_dataset("MatouK98/math_rlvr_filtered_audited", split="train")
assert len(train) == 18106

# Equivalent named subset:
same_train = load_dataset(
    "MatouK98/math_rlvr_filtered_audited",
    "qwen3_4b_correct_0_2",
    split="train",
)

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