Datasets:

jinyuan222
/

R-Quest-Validity-RL

answer_verified bool 1 class	canonical_final_answer stringlengths 1 912 ⌀	id stringlengths 18 18	invalid_type stringclasses 7 values	question stringlengths 45 1.67k	round stringclasses 5 values	split stringclasses 1 value	terra_label stringclasses 6 values	terra_validity stringclasses 2 values	validity_rl_target stringlengths 1 912	answer_confidence float64 0.86 1 ⌀
null	null	q_0015e31978620fd6	ambiguity	In the land of Puzzlia, the wise mathematician, Professor Numble, has devised a game to challenge the young minds of the kingdom. The game board is an infinite grid of squares, with each square numbered according to its row and column, starting from (1,1). The game begins with a piece placed at the origin (1,1). In eac...	v1	train	C	INVALID	INVALID	null
true	673	q_00fd275e36483334	null	What is the smallest positive integer n such that $n! \equiv 0 \pmod{2019}$? (Hint: Consider the prime factorization of 2019 and the nature of factorials.)	v1	train	A	VALID	673	1
true	By the Cauchy–Schwarz inequality, \[ \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge \frac{(a+b+c)^2}{a+b+c}=\frac{9}{3}=3. \] Equality holds when $a/b=b/c=c/a$, which for positive $a,b,c$ gives $a=b=c=1$.	q_011e41cb067fbbe3	null	Given positive real numbers $ a, b, c $ such that $ a + b + c = 3 $, prove that: \[ \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq 3. \]	v1	train	A	VALID	By the Cauchy–Schwarz inequality, \[ \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge \frac{(a+b+c)^2}{a+b+c}=\frac{9}{3}=3. \] Equality holds when $a/b=b/c=c/a$, which for positive $a,b,c$ gives $a=b=c=1$.	1
null	null	q_012eadad9fc30481	ambiguity	In a futuristic city, every resident is either a truth-teller (who always tells the truth) or a liar (who always lies). One day, a curious young mathematician decides to investigate the city's population by asking a series of questions to various residents. He picks a random resident and asks, "Is your friend a truth-t...	v5	train	C	INVALID	INVALID	null
null	null	q_01679b5f9f55c7b3	no_solution	{Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 10$ and $P(2) = 20$. Prove that there is no integer $k$ for which $P(k) = 30$, given that $P(x)$ has degree at least 3.}	v4	train	D	INVALID	INVALID	null
true	260 square feet	q_0184b2f2c390fddb	null	A rectangular garden is divided into four smaller rectangles by two straight paths, one horizontal and one vertical. The paths are 2 feet wide and run the entire length of the garden. If the total area of the garden (including the paths) is 320 square feet, and the total area of the paths alone is 60 square feet, what ...	v1	train	A	VALID	260 square feet	1
true	0	q_0196ca4d4887c047	null	In the complex plane, let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. For how many complex numbers $z$ is the product of the real part of $z$ and the imaginary part of $z$ a positive integer and $P(z) = 0$?	v4	train	A	VALID	0	1
null	null	q_01a6ea1848beba46	contradiction	A circle is inscribed in an isosceles triangle with sides of lengths 13, 14, and 15 units. A smaller circle is then inscribed in the same triangle, tangent to the two equal sides and to the inscribed circle. Find the radius of the smaller circle.	v3	train	D	INVALID	INVALID	null
null	null	q_01b0265bfc050b97	malformed	A convex polygon $ P $ in the plane has vertices $ V_1, V_2, \ldots, V_n $ and an area $ A(P) $. Let $ C $ be the circumcircle of $ P $ and $ r $ be its radius. Consider the function $ f(x) = \frac{A(P + xC)}{A(P)} $, where $ P + xC $ denotes the polygon obtained by scaling $ C $ by a factor of \( x \...	v4	train	E	INVALID	INVALID	null
null	null	q_01c3d3fa860fbf9f	no_solution	A magical garden is arranged in a perfect circle and contains a variety of flowers. Each flower blooms only on the days when the total number of blooms is a perfect square. Given that on day 1, there is exactly 1 blooming flower, and the number of blooming flowers doubles every 2 days, find the smallest day number wher...	v5	train	D	INVALID	INVALID	null
null	null	q_01d178a6b13fc041	ambiguity	Given a regular 2021-gon $P$ inscribed in a circle with radius $r$, let $A$, $B$, and $C$ be three vertices of $P$ such that the arc length between $A$ and $B$, $B$ and $C$, and $C$ and $A$ are in arithmetic progression. Determine the area of the triangle formed by $A$, $B$, and $C$ in ter...	v3	train	C	INVALID	INVALID	null
true	4950	q_01de192facf3c9be	null	In a country with 100 cities, every city is connected to every other city by a road. Each road is uniquely colored and no two roads of the same color share an endpoint. What is the maximum number of colors that can be used for the roads under these conditions?	v1	train	A	VALID	4950	0.99
true	3	q_01e6b354801d98da	null	What is the smallest positive integer $ n $ such that the expression $ \frac{n^2 + 2n + 1}{n^2 - 3n + 2} $ is an integer?	v1	train	A	VALID	3	1
null	null	q_01f12032618e2caf	malformed	In a sequence of positive integers, each term after the first is obtained by adding the previous two terms. The first term is 1, and the second term is a positive integer less than 10. If the 10th term in this sequence is a multiple of 12, find the number of possible values for the second term.	v3	train	E	INVALID	INVALID	null
true	24	q_01fb654a0317452b	null	In the kingdom of Numeralia, there is a unique type of gemstone known as the "Prime Orb." Each Orb contains a prime number, $ p $, and the King has decided to distribute these Orbs among his 12 most distinguished scholars. Each scholar can only receive one Orb, and each Orb's prime number must be a divisor of the sum...	v4	train	A	VALID	24	0.99
null	null	q_0207e2abeb032c4f	contradiction	Let $ P $ be a convex polygon with $ n $ sides. A beautiful coloring of $ P $ is a coloring of the vertices and edges of $ P $ using exactly three colors such that: 1. Any two adjacent vertices (connected by an edge) have different colors. 2. Any two vertices on the same side (not necessarily adjacent) have ...	v5	train	D	INVALID	INVALID	null
true	1	q_0234f84ac6a82a9e	null	Find the smallest positive integer n such that the sum of the cubes of the first n positive integers is divisible by the product of the first n positive integers.	v2	train	A	VALID	1	1
true	$\frac{51017}{81642}$	q_0235235c1cfb4de0	null	在平面直角坐标系中，有一系列由正整数组成的点集合 $ S = \{ (x, y) \| x, y \in \mathbb{Z}^+, x + y \leq 100 \} $，其中 $ \mathbb{Z}^+ $ 表示所有正整数的集合。现在从集合 $ S $ 中随机选择三个不同的点 $ A(x_1, y_1) $, $ B(x_2, y_2) $ 和 $ C(x_3, y_3) $。设这三个点组成的三角形面积为 $ T $。求 $ T $ 为整数的概率。	v2	train	A	VALID	$\frac{51017}{81642}$	0.99
true	101	q_026d6575590f2df3	null	In the Cartesian coordinate system, let $P$ be a point with integer coordinates such that $1 \leq P_x, P_y \leq 100$. A path from $P$ to another point $Q$ is defined as a sequence of steps, each moving either one unit to the right or one unit up, without revisiting any previously visited point. A special path is a ...	v5	train	A	VALID	101	0.99
true	8	q_0271acef0bec7b0b	null	How many positive integers less than 500 are divisible by 4, 5, and 6?	v1	train	A	VALID	8	1
null	null	q_02ab066130fe57ed	no_solution	在三维空间中，有三个正交坐标轴 $x, y, z$，轴上分别标记了点 $A_1, A_2$ 和 $B_1, B_2$。$A_1$ 的坐标为 $(a_1, 0, 0)$，$A_2$ 的坐标为 $(0, a_2, 0)$，而 $B_1$ 和 $B_2$ 的坐标分别为 $(b_1, 0, 0)$ 和 $(0, b_2, 0)$，其中 $a_1, a_2, b_1,$ 和 $b_2$ 是互不相同的正整数。如果从原点 $O$ 出发，沿着正交路径到达 $A_1$，然后到达 $B_1$，接着返回原点，再沿另一条不同的路径到达 $A_2$，最后到达 $B_2$，已知整个行程中经...	v4	train	D	INVALID	INVALID	null
null	null	q_02bee8e9f6f7bba0	contradiction	Let $ P(x) $ be a polynomial with integer coefficients such that $ P(1) = 2023 $ and $ P(2) = 4046 $. Suppose that there exists an integer $ n $ for which $ P(n) = 2023 $ and $ P(n+1) = 2024 $. Find the smallest possible value of $ n $. For example, if $ P(x) = x^2 + 1903 $, then $ P(1) = 2023 $ and ...	v3	train	D	INVALID	INVALID	null
null	null	q_02ca020b49db8f21	ambiguity	In a magical forest, there are trees that grow in a unique pattern. Each tree has a certain number of branches, and each branch can grow in one of four directions: north, south, east, or west. However, no two branches on the same tree can grow in the same direction. Trees are magically programmed to grow their branches...	v5	train	C	INVALID	INVALID	null
null	null	q_02e72163e0a02fca	contradiction	Find all positive integers $ n $ such that the equation $ x^2 + y^2 + z^2 + 3(x + y + z) = nxyz $ has a solution in positive integers, and additionally, satisfy the constraints: 1. $ x, y, z \leq 10 $ 2. $ n < 100 $ 3. $ x, y, z $ are pairwise coprime. Which of the following is a possible value of $ n $ if...	v3	train	D	INVALID	INVALID	null
null	null	q_0304b0789e3452cf	no_solution	Let $f(n)$ be the number of ways to express $n$ as the sum of positive integers, where the order of the summands matters. For example, $f(3) = 4$ because $3$ can be expressed as $3, 2+1, 1+2,$ and $1+1+1$. Find the smallest positive integer $n$ such that $f(n) = 120$.	v1	train	D	INVALID	INVALID	null
null	null	q_032dc211ecbc1d01	malformed	在平面直角坐标系中，给定两个点 $A(1, 2)$ 和 $B(4, 7)$，以及一条圆周 $C: (x-3)^2 + (y-3)^2 = 4$。设 $P$ 是圆周 $C$ 上任意一点，直线 $AP$ 和直线 $BP$ 分别与圆周 $C$ 在不同于 $P$ 的点处相交，记为 $Q_A$ 和 $Q_B$ 分别对应。令 $L$ 为直线 $PQ_A$ 和直线 $PQ_B$ 之比的最小值。试求圆周 $C$ 对直线 $AB$ 沿着 $P$ 的$L$的最大可能值。	v4	train	E	INVALID	INVALID	null
null	null	q_0375dd1d98024ab8	malformed	In the mystical land of Numerica, there exists an ancient city called Algoradda. The city is organized into $n$ concentric rings, with the innermost ring being Ring 1, and the outermost ring being Ring $n$. Each ring is inhabited by a different species of creatures, and the number of creatures in each ring forms a uniq...	v5	train	E	INVALID	INVALID	null
true	$\frac{\pi}{3\sqrt{3}}$	q_03872f070570d349	null	A circle is inscribed in an equilateral triangle with side length $ s $. If a point is randomly selected from inside the triangle, what is the probability that it lies within the circle?	v1	train	A	VALID	$\frac{\pi}{3\sqrt{3}}$	1
null	null	q_038c52710b7fd627	no_solution	In a mysterious ancient city, there are $N$ magical stones, each one endowed with a unique power level that is a positive integer. Legend has it that when a person seeks power, they must traverse a mystical path consisting of $N$ points connected by exactly $N-1$ unidirectional roads, forming a tree structure. The powe...	v3	train	D	INVALID	INVALID	null
null	null	q_039cc2d7e2b2ab97	no_solution	Consider a sequence of integers $(a_n)$ defined by $a_1 = 1$, $a_2 = 2$, and for $n \geq 3$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that the polynomial $P(x) = x^3 + a_n x^2 + a_{n-1} x + a_{n-2}$ has three distinct real roots. Find the value of $a_{10}$.	v2	train	D	INVALID	INVALID	null
null	null	q_03a0e8911f0f5f56	contradiction	Let $ p(x) $ be a monic polynomial of degree $ n $ with real coefficients such that $ p(0) = 1 $ and for all real numbers $ x $, the equation \[ p(x) + p(1 - x) = 2x^2 + 1 \] holds. Determine the sum of all roots of the equation $ p(x) = x $.	v4	train	D	INVALID	INVALID	null
null	null	q_03ccc7eb0825916b	ambiguity	Find all integers $ n $ such that the equation \[ \sum_{k=1}^{n} \frac{1}{k^2} = \frac{a}{b} \] has integer solutions for $ a $ and $ b $ with $ \gcd(a, b) = 1 $, where $ b \neq 1 $. How many such $ n $ exist, and what are the corresponding fractions $ \frac{a}{b} $?	v3	train	C	INVALID	INVALID	null
null	null	q_03e72870211638a7	no_solution	In the complex plane, consider the function $ f(z) = z^3 - 3z^2 + 2z $. Let $ \gamma $ be the circle $ \|z\| = 2 $ traversed counterclockwise. Using the argument principle, determine the number of zeros of $ f(z) $ inside $ \gamma $, counting multiplicities.	v2	train	D	INVALID	INVALID	null
null	null	q_03f41e2385ffefe8	contradiction	In the mystical land of Mathoria, there exists a magical sequence of numbers, the Mathorian Harmonies. The sequence is defined by the recurrence relation: $a_{n+2} = a_{n+1} + a_n + a_{n-1}$ for all $n \geq 2$, with initial conditions $a_1 = 1$, $a_2 = 1$, and $a_3 = 2$. Each term of the sequence is also as...	v5	train	D	INVALID	INVALID	null
true	448	q_04397cd08208d5fd	null	Let $S$ be the set of all positive integers $n$ such that $n^2 + 12n - 2007$ is a perfect square. Find the sum of all elements in $S$ that are less than 1000.	v2	train	A	VALID	448	1
true	128 square units	q_043d80987e42e7dc	null	Three distinct points $ A $, $ B $, and $ C $ lie on a circle of radius $ 10 $ units. If the distance from the center of the circle to the midpoint of segment $ AC $ is $ 6 $ units, what is the maximum possible area of triangle $ ABC $?	v1	train	A	VALID	128 square units	1
true	1	q_0440e3c6e82c8727	null	Find the smallest positive integer $ n $ such that the number of integers $ k $ satisfying $ 1 \leq k \leq n $ and $ k^3 \equiv 1 \pmod{n} $ is equal to the number of integers $ m $ satisfying $ 1 \leq m \leq n $ and $ m^2 \equiv 1 \pmod{n} $.	v2	train	A	VALID	1	0.99
null	null	q_044361e979849562	contradiction	A convex polyhedron has 18 vertices and 32 faces. Determine the number of edges of the polyhedron, knowing that each vertex is connected to 5 other vertices, and every face is a triangle or a quadrilateral. Additionally, if one edge is removed, how many triangles remain?	v3	train	D	INVALID	INVALID	null
true	24	q_04447f9941be7e23	null	What is the remainder when the product of the first 100 positive integers that are not divisible by 5 is divided by 100?	v1	train	A	VALID	24	0.99
null	null	q_045396c257c17aa6	contradiction	In the coordinate plane, a square $ABCD$ has vertices $A(0,0)$, $B(0,6)$, $C(6,6)$, and $D(6,0)$. A point $P$ is chosen inside the square such that the distances from $P$ to the sides of the square are $p$, $q$, $r$, and $s$ where $p + q = r + s = 3$. Find the area of quadrilateral $APCB$.	v1	train	D	INVALID	INVALID	null
null	null	q_0463804b11e40d77	no_solution	There exists a sequence of real numbers $a_1, a_2, \ldots, a_n$ where $n \geq 2$. The sequence has the property that for each $k$ (with $1 \leq k < n$), $a_{k+1}$ is the average of the previous $k$ terms. Given that $a_1 = 1$ and the sum of all terms in the sequence is equal to $n + \frac{1}{2}$, find t...	v2	train	D	INVALID	INVALID	null
null	null	q_04682d95fad0f1ff	contradiction	Let $ f(x) $ be a polynomial of degree 5 with integer coefficients such that $ f(1) = 0 $ and $ f(2) = 10 $. If $ f(x) $ has exactly one real root $ r $ in the interval $ (1, 2) $ and all other roots are non-real, find the sum of the coefficients of $ f(x) $.	v1	train	D	INVALID	INVALID	null
true	5050	q_046f7a7da54ba4af	null	Let $ f $ be a function defined on the set of non-negative integers with the following properties: 1. $ f(1) = 1 $ 2. $ f(a + b) = f(a) + f(b) + ab $ for all non-negative integers $ a $ and $ b $. Find the value of $ f(100) $.	v1	train	A	VALID	5050	1
true	Yes; the minimum total tax is 140500 gold coins.	q_0475cd7753fd15c7	null	In the kingdom of Numeralia, there are 1000 villages, each with a unique number from 1 to 1000. The king of Numeralia has issued a decree that every two villages are connected by a direct road if and only if the numbers of the two villages are relatively prime. A tourist, keen on exploring the kingdom, starts in villag...	v5	train	A	VALID	Yes; the minimum total tax is 140500 gold coins.	0.99
null	null	q_04b480470656a454	ambiguity	In a fictional kingdom, there are 20 cities connected by a network of roads such that every pair of cities is directly linked by exactly one road. A traveler wishes to visit each city exactly once, starting and ending at the same city. However, due to a peculiar law, the traveler cannot traverse the same road twice. Ho...	v3	train	C	INVALID	INVALID	null
true	7 integers; sum = 820	q_04d40aacb5898f02	null	In a particular infinite sequence of integers, each term is defined recursively by the formula $ a_{n} = 2a_{n-1} + 3a_{n-2} $ with initial conditions $ a_{0} = 1 $ and $ a_{1} = 2 $. Determine the number of integers between 1 and 1000 inclusive that can be expressed as a term in this sequence. Additionally, find...	v5	train	A	VALID	7 integers; sum = 820	1
null	null	q_0504bd28d08f48a7	ambiguity	In a hyperbolic plane, consider a regular polygon with $n$ sides, where each internal angle is $\alpha$. If the polygon is inscribed in a circle of radius $r$ in this hyperbolic plane, find the relationship between $n$, $\alpha$, and $r$. Specifically, derive a formula for $n$ in terms of $\alpha$ and \...	v4	train	C	INVALID	INVALID	null
null	null	q_053d2fc7701c5a85	contradiction	Find all real numbers $ x $ such that the sequence $ \{a_n\} $ defined by $ a_1 = x $ and $ a_{n+1} = a_n^2 - 2 $ for $ n \geq 1 $ converges to a limit $ L $ where $ L > 0 $ and $ L^2 = 2L + 1 $.	v3	train	D	INVALID	INVALID	null
true	1	q_055d3b97c9e3a46e	null	In the mystical land of Numeralia, a wise wizard has enchanted a magical forest with three types of trees: Aria Trees, Bora Trees, and Ciel Trees. The wizard has set a peculiar rule: every Aria Tree must be at least twice as tall as any Bora Tree, and every Bora Tree must be at least three times as tall as any Ciel Tre...	v3	train	A	VALID	1	0.99
null	null	q_058d698ad855a642	malformed	Let $ S $ be the set of all permutations of the set $ \{1, 2, 3, \ldots, 12\} $. For each permutation $ \sigma \in S $, let $ \sigma(n) $ denote the position of the number $ n $ in the permutation. Determine the number of permutations $ \sigma $ such that for every $ n $ and $ k $ in the set \( \{1, 2, ...	v3	train	E	INVALID	INVALID	null
true	n \in \{2,3,4,7\}	q_058f40b6b665501d	null	Find all positive integers $ n $ such that $ n^2 + 3n + 2 $ is divisible by $ n - 1 $.	v1	train	A	VALID	n \in \{2,3,4,7\}	1
null	null	q_0594e8f9f81730bd	malformed	In the complex plane, let $z_1, z_2, \ldots, z_n$ be $n$ distinct points such that the sum of any two points, when their product is taken modulo the prime number $p = 2023$, results in a perfect square. If $n = 2022$, determine the largest possible value of the expression: \[ \sum_{1 \le i < j \le 2022} \left\| \frac{z_...	v4	train	E	INVALID	INVALID	null
null	null	q_05a8e02957fc54de	contradiction	In the mystical land of Mathemoria, the inhabitants have a unique way of celebrating the annual Unity Festival. They create a large, symmetrical pyramid of stones, where each stone is uniquely numbered from 1 to N (where N is an odd number). The stones are arranged in such a way that the base of the pyramid is a square...	v5	train	D	INVALID	INVALID	null
true	\frac{1}{3}	q_05bbf725d26ee45e	null	Let $ S $ be the set of all non-degenerate triangles with integer side lengths and a perimeter of 12. Determine the number of such triangles $ S $ that have an area that is an integer, and express your answer as a fraction of the total number of triangles in $ S $. Given the condition that the side lengths \( a,...	v2	train	A	VALID	\frac{1}{3}	1
true	1	q_05c383bf0cd2be83	null	What is the smallest positive integer $ n $ such that the sum of the squares of the first $ n $ positive integers is a perfect cube?	v1	train	A	VALID	1	1
true	No positive integers $n$.	q_05c6c2c1e9b60936	null	Find all positive integers $ n $ such that $ 2^n + 3^n + 6^n + 7^n $ is a perfect square.	v2	train	A	VALID	No positive integers $n$.	1
true	42	q_05dce082506413d2	null	In a unique calendrical system, each year is divided into cycles of $13$ months, where each month has $28$ days. Suppose a traditional festival is observed every $3$ years in the current year. If the festival was held in year $0$ and it has been observed on $15$ distinct years, what is the smallest possible c...	v3	train	A	VALID	42	0.99
null	null	q_060f26a00aae4e71	contradiction	In the realm of abstract geometry, consider a tetrahedral complex $T$ embedded in a three-dimensional space. $T$ is composed of $n$ tetrahedra, each defined by four non-coplanar points. Each edge of the tetrahedra is colored either red or blue, with no two adjacent edges sharing the same color. Moreover, no three...	v4	train	D	INVALID	INVALID	null
null	null	q_066e14ae8a652222	no_solution	Find the smallest positive integer $ n $ such that $ 3^n $ and $ 5^n $ both have exactly 2024 digits when expressed in base 10. Use this information to determine the last three digits of $ 15^n $.	v3	train	D	INVALID	INVALID	null
null	null	q_06832bef59da9713	missing_information	In a game where two players, A and B, take turns to remove one or two stones from a pile of n stones, the player who takes the last stone wins. However, if a player takes two stones, the other player must also take two stones on their next turn if possible. For which values of n between 1 and 100 inclusive can player A...	v3	train	C	INVALID	INVALID	null
null	null	q_06fbc1fa29d87213	no_solution	Let $ABC$ be a triangle with circumcircle $\omega$. Let $D$ be a point on the arc $BC$ not containing $A$ in $\omega$. The tangents to $\omega$ at $B$ and $D$ intersect at $E$. If $P$ and $Q$ are the feet of the perpendiculars from $A$ to $BD$ and $CE$ respectively, prove that $\angle APQ = \angle AQP$.	v1	train	D	INVALID	INVALID	null
null	null	q_070a91a31ab26443	contradiction	A finite sequence of non-zero digits is formed by concatenating all positive integers in increasing order, i.e., "123456789101112..." Continue this sequence infinitely. What is the 1000th digit of this sequence?	v2	train	D	INVALID	INVALID	null
null	null	q_070e9d0d30f3b138	ambiguity	在一个无限大的网格棋盘上，每个格子可以被标记为黑或白。游戏开始时，从一个角落开始，黑白棋子交错地摆放在相邻的格子上，形成一种特殊的模式。游戏的目标是通过一系列合法操作尽可能多地改变棋盘的颜色。合法操作是指选择一条直线，将这条直线上所有的棋子颜色反转（即，如果一个格子原本是黑色，则变为白色；如果是白色，则变为黑色）。具体规则如下： 1. 操作只能针对整个行或列，不能只针对单个格子。 2. 如果一个棋盘被操作后所有的棋子颜色都相同，则该棋盘被视为“成功”，并停止操作。 3. 为了增加游戏的趣味性和挑战性，定义一个“翻转数”（Flip Number）——这是指执行一次操作所需的棋子个数。例如，在一个 3x3 的棋盘上，翻转整行需要翻转...	v4	train	C	INVALID	INVALID	null
null	null	q_073ebba5ed6e340b	ambiguity	Let $ S $ be a set of $ n $ distinct positive integers, where $ n \geq 3 $. Define a function $ f: \mathcal{P}(S) \to \mathbb{Z} $ on the power set of $ S $ such that for any subset $ A \subseteq S $, $ f(A) $ is the number of ways to partition $ A $ into two non-empty subsets such that the sum of the e...	v4	train	C	INVALID	INVALID	null
null	null	q_07df57ba8647b728	contradiction	A regular octahedron is inscribed within a sphere such that all its vertices touch the sphere. Each edge of the octahedron is 1 unit long. A light source is placed at the center of the sphere, casting shadows of the octahedron onto a flat wall directly opposite the octahedron's vertex. What is the area of the shadow ca...	v5	train	D	INVALID	INVALID	null
null	null	q_07e5f5267175bc51	ambiguity	Three mathematicians, Alice, Bob, and Carol, each have a distinct positive integer on their foreheads, and they see each other's numbers. They are given the sum of the squares of their numbers but not the numbers themselves. They must simultaneously guess whether their own number is larger than or smaller than the aver...	v5	train	C	INVALID	INVALID	null
true	Exactly the integers whose decimal digits are obtained as follows: choose a finite list $a_1,\ldots,a_r\in\{2,\ldots,9\}$, possibly empty, and include exactly $m=\prod_{i=1}^r a_i-\sum_{i=1}^r a_i$ copies of the digit $1$, in any order. Here the empty product is $1$ and empty sum is $0$, giving $n=1$. No ze...	q_07fb384dd53e5418	null	Find all positive integers $ n $ such that $ n $ divides the product of its digits minus the sum of its digits, i.e., $ n \mid (d_1 \cdot d_2 \cdot \ldots \cdot d_k - (d_1 + d_2 + \ldots + d_k)) $, where $ d_1, d_2, \ldots, d_k $ are the digits of $ n $.	v2	train	A	VALID	Exactly the integers whose decimal digits are obtained as follows: choose a finite list $a_1,\ldots,a_r\in\{2,\ldots,9\}$, possibly empty, and include exactly $m=\prod_{i=1}^r a_i-\sum_{i=1}^r a_i$ copies of the digit $1$, in any order. Here the empty product is $1$ and empty sum is $0$, giving $n=1$. No ze...	0.99
null	null	q_08454bb1ce81845f	ambiguity	Consider a regular hexagon $ABCDEF$ with side length $10$. Each vertex is connected to every other vertex by a diagonal. Find the number of diagonals that intersect the interior of the hexagon, passing through more than two vertices. Additionally, determine the length of the longest diagonal that does not pass through ...	v2	train	C	INVALID	INVALID	null
null	null	q_086f92c0e1b4f9b1	no_solution	In the mystical land of Numeria, a peculiar tree grows called the "Tripletizer," whose fruits reveal a number when peeled. A wise sage, intrigued by the patterns in the numbers revealed, noticed that the Tripletizer's fruits displayed numbers in a sequence that adhered to the following rule: the $n^{th}$ fruit's number...	v5	train	D	INVALID	INVALID	null
true	10^\circ	q_0883e93bd725021a	null	In an isosceles triangle $ABC$ with $AB = AC$ and $\angle BAC = 20^\circ$, point $D$ is on $AB$ such that $AD = BC$. Additionally, point $E$ is on $AC$ such that $AE = BD$. Find the measure of $\angle ACD$.	v1	train	A	VALID	10^\circ	0.999
true	6	q_08c9a250498074ba	null	A positive integer $ n $ is called super-decomposable if $ n $ can be written as $ n = ab $, where $ a $ and $ b $ are integers greater than 1 such that $ \gcd(a, b) = 1 $. What is the smallest super-decomposable number?	v1	train	A	VALID	6	1
true	5508	q_08dd7d9695e709f1	null	In a complex hexagonal lattice, each node is connected to six neighboring nodes by edges. A particle starts at the center node and moves to one of its neighbors with equal probability at each step. It is known that the particle can revisit nodes it has already visited. How many distinct paths of exactly 5 steps can the...	v5	train	A	VALID	5508	0.99
null	null	q_091558d2240ba2c1	ambiguity	Let $P(x)$ be a polynomial with integer coefficients such that $P(10) = 1000$ and $P(100) = 100000$. Given that $P(x)$ has a degree of at most $5$, find the number of possible integer roots that $P(x)$ can have.	v2	train	C	INVALID	INVALID	null
true	541	q_09343e18de338536	null	A sequence of positive integers $a_1, a_2, a_3, \ldots$ is defined recursively by $a_1 = 2$, and for $n \geq 2$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that no term $a_k$ (for $1 \leq k < n$) divides $a_n$. Find the remainder when $a_{100}$ is divided by 1000.	v1	train	A	VALID	541	1
true	990	q_098f566736dc6e57	null	In the plane, consider a set of $2021$ distinct points, no three of which are collinear. These points represent the vertices of a convex polygon. A "score" of a pair of points is defined as the number of triangles formed by these points with both vertices as vertices of one of the triangles. If the total score for all ...	v5	train	A	VALID	990	0.99
null	null	q_099b39b58307c919	no_solution	What is the smallest positive integer $n$ such that for all integers $a$ and $b$ where $1 \leq a < b < n$, the product $ab$ is not divisible by $n$, given that $n$ is the product of two distinct primes, $n = pq$ where $p$ and $q$ are prime numbers?	v4	train	D	INVALID	INVALID	null
true	n=9	q_09dee11b1ad72ee3	null	Find all positive integers $ n $ for which $ n^3 - 14n^2 + 53n - 72 $ is a perfect square.	v2	train	A	VALID	n=9	0.99
true	2023	q_09fa0495f139c00e	null	A sequence of positive integers $a_1, a_2, a_3, \ldots, a_n$ is defined as follows: $a_1 = 1$, $a_2 = 2$, and for $n \geq 3$, $a_n$ is the smallest positive integer not among $a_1, a_2, \ldots, a_{n-1}$ such that $a_n$ is relatively prime to $a_{n-1}$. Determine the value of $a_{2023}$.	v1	train	A	VALID	2023	1
true	1/1	q_0a1251cf977abf69	null	Find the number of ordered pairs of positive integers $(m,n)$ such that the sum of the squares of $m$ and $n$ equals the sum of the cubes of $m$ and $n$. Express your answer as a common fraction.	v1	train	A	VALID	1/1	1
null	null	q_0a6f111ac5b78bb9	contradiction	Let $ f(x) $ be a polynomial of degree 3 such that $ f(1) = 1 $, $ f(2) = 4 $, $ f(3) = 9 $, and $ f(4) = 16 $. Find the value of $ f(5) $.	v1	train	D	INVALID	INVALID	null
null	null	q_0aa39f15e0684a2d	ambiguity	在三维欧几里得空间中，考虑所有单位向量 $\mathbf{u} = (x, y, z)$ 满足条件 $\|x\| + \|y\| + \|z\| \leq 1$。设 $S$ 为这些单位向量的集合。现在，从 $S$ 中随机选取一个点 $\mathbf{p}$，然后沿着从原点到 $\mathbf{p}$ 的直线方向飞行，飞行的距离等于从原点到 $\mathbf{p}$ 的距离。求在飞行过程中，与初始位置相距不超过单位距离的所有可能终点的体积，表示为 $V$。设 $V$ 的值为 $A\pi + B\sqrt{C} + D\sqrt{E}$，其中 $A, B, C, D, E$ 是整数，且 $C$ 和 $E$ 无平方因子。求 $A + B + C + ...	v5	train	C	INVALID	INVALID	null
true	Countably infinitely many	q_0ab1a56491e29270	null	Find the number of ordered pairs $(a, b)$ of integers such that the equation \[ x^2 + ax + b = 0 \] has integer roots and the sum of the roots is a multiple of 10.	v1	train	A	VALID	Countably infinitely many	1
true	2	q_0ab6e4b0143a3a45	null	Find the number of real solutions to the equation $ x^{10} + 9x^9 + 14x^8 + 1 = 0 $.	v1	train	A	VALID	2	0.99
null	null	q_0ab93b1ca1d4909f	ambiguity	在无限大的棋盘上，有n个棋子，每个棋子的位置由整数坐标(x,y)表示。这些棋子之间的曼哈顿距离（\|x1-x2\| + \|y1-y2\|）大于等于k。给定n和k，请设计一个算法来确定是否可以将这n个棋子放置在一个方形网格（边长为整数）内，并且这个网格满足所有棋子之间的最小曼哈顿距离至少为k。如果可以找到这样的网格，请返回网格的最小边长；如果不能，返回-1。	v5	train	C	INVALID	INVALID	null
null	null	q_0abe71a23db1c2fa	contradiction	In a mystical village, there exists a tree with a unique property: it grows fruits in such a way that the number of fruits in each generation follows a specific mathematical pattern. The tree starts with 1 fruit in the first generation. In the second generation, it doubles the number of fruits to 2. From the third gene...	v5	train	D	INVALID	INVALID	null
null	null	q_0b01a890d5a3c073	no_solution	In a peculiar country, there exists a set of $n$ cities, each connected by a unique bidirectional road to every other city. Each road has an associated cost, and it's known that the total sum of all road costs is divisible by 7. A traveler wishes to visit all $n$ cities exactly once and return to the starting city,...	v5	train	D	INVALID	INVALID	null
null	null	q_0b6d79b4c821edf9	contradiction	In the Cartesian plane, a particle moves from the origin $ (0, 0) $ to the point $ (n, n) $ by taking unit steps either right or up, and then from $ (n, n) $ to $ (2n, 0) $ by taking unit steps either left or down. Let $ P_n $ be the number of distinct paths the particle can take. Prove that $ P_n $ is divi...	v2	train	D	INVALID	INVALID	null
true	4\sqrt{2}	q_0b76e8eb7817eb69	null	Let $f(x) = x^4 - 4x^3 + 10x^2 - 12x + 9$ be a polynomial with integer coefficients. Suppose that $r_1, r_2, r_3, r_4$ are the roots of $f(x)$, not necessarily distinct, in the complex plane. Find the smallest possible value of $$ \|r_1 - r_2\| + \|r_2 - r_3\| + \|r_3 - r_4\| + \|r_4 - r_1\| $$ where $\| \cdot \|$ denotes the co...	v4	train	A	VALID	4\sqrt{2}	0.99
null	null	q_0b861bfab252d432	ambiguity	在平面上有 $n$ 条互不相同的直线，每条直线将平面分为两个半平面。这些直线最多能将平面分成多少个区域？已知 $n \geq 3$，且任意三条直线不共点。此外，给出任意一条直线和所有其他直线交点的坐标，请求出所有区域的面积之和。	v4	train	C	INVALID	INVALID	null
null	null	q_0b948e8660937e56	no_solution	In a unique coordinate system on a sphere of radius $R$, every point on the surface is defined by its spherical coordinates $(\theta, \phi)$, where $\theta$ is the polar angle (ranging from $0$ to $\pi$) and $\phi$ is the azimuthal angle (ranging from $0$ to $2\pi$). Consider a set of $n$ points on th...	v3	train	D	INVALID	INVALID	null
true	0	q_0bb21cfe1d36e0d4	null	In the sequence $ a_n $ defined by $ a_1 = 1 $ and $ a_{n+1} = a_n + \frac{1}{a_n} $ for all $ n \geq 1 $, determine the integer part of the sum $ \sum_{k=1}^{100} \left( \frac{1}{a_k + a_{k+1}} \right)^2 $.	v2	train	A	VALID	0	0.99
true	Infinitely many (all integers k \ge 0)	q_0bcf3de40388ac9b	null	In a sequence of positive integers $a_1, a_2, \ldots, a_{2023}$, each term $a_n$ satisfies $a_n = n^2 + k$ for some fixed integer $k$. It is known that for each $n$, $a_n$ divides the product of the first $n$ terms of the sequence, $a_1 a_2 \cdots a_n$. Find the number of distinct possible values of \(k...	v2	train	A	VALID	Infinitely many (all integers k \ge 0)	1
null	null	q_0befc6a3c3d6cf65	missing_information	In the Cartesian plane, consider a circle centered at the origin with radius 1. Two distinct points $A$ and $B$ lie on this circle such that the line segment $AB$ subtends an angle $\theta$ at the center of the circle, where $0 < \theta < \pi$. If the product of the $y$-coordinates of $A$ and $B$ is equ...	v1	train	C	INVALID	INVALID	null
true	Take c=2ab+1=ab+(ab+1). Then c∈S, while c≡1 (mod a) and c≡1 (mod b), so neither a nor b divides c.	q_0c49af70e867e2e2	null	Let $ S $ be the set of all positive integers that can be represented as the sum of two or more consecutive positive integers, starting from some integer $ n \geq 1 $. For example, $ 9 $ is in $ S $ because it can be written as $ 2 + 3 + 4 $. Prove that if $ a $ and $ b $ are two distinct elements of \( S...	v3	train	A	VALID	Take c=2ab+1=ab+(ab+1). Then c∈S, while c≡1 (mod a) and c≡1 (mod b), so neither a nor b divides c.	1
null	null	q_0c6db88f38b2af2d	no_solution	A sequence of positive integers is defined by $ a_1 = 1 $, $ a_2 = 2 $, and for $ n \geq 3 $, $ a_n = a_{n-1} + a_{n-2} + d_n $, where $ d_n $ is the smallest positive integer not appearing earlier in the sequence that is also not a multiple of any previous $ a_i $. Find $ a_{10} $.	v2	train	D	INVALID	INVALID	null
null	null	q_0c743595f63c5f50	missing_information	In a small village, there are 100 families. Each family has either 1, 2, or 3 children. The total number of children in the village is exactly 200. If each family with 2 children has exactly 1 dog, and each family with 3 children has exactly 2 dogs, how many dogs are there in the village?	v1	train	C	INVALID	INVALID	null
true	198	q_0c99039330c6eb97	null	Find the number of ordered pairs $(a, b)$ of positive integers such that $a, b \leq 100$ and $ab = \left\lfloor \frac{a}{b} \right\rfloor + \left\lfloor \frac{b}{a} \right\rfloor$.	v2	train	A	VALID	198	1
true	1	q_0cbb36c23537a5d1	null	Find the smallest positive integer $ n $ such that $ 2^n + n $ is divisible by $ n $.	v2	train	A	VALID	1	1
true	189	q_0ccff4ef912533e3	null	Let $ S $ be the set of all positive integers that are multiples of 3 but not of 5. Define a sequence $ \{a_n\} $ such that $ a_n $ is the smallest element in $ S $ that can be expressed as the sum of $ n $ distinct elements from $ S $. Find $ a_{10} $.	v2	train	A	VALID	189	0.99
true	1633	q_0ce928e9cd6f75c6	null	Let $S$ be a set of 100 points in the plane such that no three points are collinear and the distance between any two points is at least 1 unit. A subset $T$ of $S$ is called a "triangulation" if $T$ consists of exactly 99 points and any two points in $T$ can be connected by a straight line segment that does not pass th...	v4	train	A	VALID	1633	0.99

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R-Zero Validity-RL Terra Clean v1

Conservative clean Terra validity dataset used for Solver initialization in R-Quest.

Paper · Code · Project Page

excluded_valid.jsonl, manifest.json, and analysis/ are audit artifacts and are not training splits.

Citation

@misc{li2026rquest,
  title={Questioning the Questions: Sustaining Self-Evolution in Reasoning Models},
  author={Jinyuan Li and Chengsong Huang and Langlin Huang and Donghong Cai and Shiping Gao and Yuyi Yang and Jiaxin Huang},
  year={2026},
  eprint={2610.04299},
  archivePrefix={arXiv},
  primaryClass={cs.LG},
  url={https://arxiv.org/abs/2610.04299}
}

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Paper for jinyuan222/R-Quest-Validity-RL

Questioning the Questions: Sustaining Self-Evolution in Reasoning Models

Paper • 2610.04299 • Published 8 days ago • 69