Datasets:
problem stringlengths 35 3.05k | answer stringlengths 1 30 | gold_type stringclasses 2
values | source stringclasses 3
values | source_row int64 10 105k ⌀ | nemotron_uuid stringlengths 36 36 | problem_sha1 stringlengths 32 32 | alt_answers listlengths 0 0 |
|---|---|---|---|---|---|---|---|
Let
\[
I = \int_{0}^{1} \frac{\ln^{2} x + \ln^{2} (1-x) - \ln x \ln (1-x)}{1 - x + x^{2}} \, dx.
\]
All logarithms are natural and the integral is taken in the improper sense. Determine the exact closed-form value of \(I\) as a combination of classical constants. | \frac{2\sqrt{3}\pi^{3}}{27} | expression | nemotron-native | null | c726869b-7ff1-4bb1-befd-7ec5fc264317 | 6effa8bc658f260cfd869c7011d600d5 | [] |
设 S 是一个 35 元集合，\mathcal{F} 是一个由一些 S 到自身的映射组成的集合. 对正整数 k，称 \mathcal{F} 具有性质 P(k)，如果对任意 x,y\in S，都存在 \mathcal{F} 中的 k 个映射f_1, f_2, \cdots, f_k（可以相同），使得f_k(\cdots(f_2(f_1(x)))\cdots)=f_k(\cdots(f_2(f_1(y)))\cdots).求最小的正整数 m，使得所有具有性质 P(2019) 的集合 \mathcal{F} 都具有性质 P(m). | 595 | numeric | dapo-math-17k | 12,701 | bd1eff54-9e2e-4218-94f5-62c2410bd593 | 45d11f0561e5509cefd2c96537d74a19 | [] |
Example 13 Let $X$ be a 56-element set. Find the smallest positive integer $n$ such that for any 15 subsets of $X$, if the union of any 7 of them has at least $n$ elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
(2006 CMO Problem) | 41 | numeric | skywork-or1 | 94,655 | 990fac00-ee86-4fba-80ae-a7b54ac4d7fb | a837de9ad2bb3a85dc19fe8e9f05d601 | [] |
在一次数学奥林匹克比赛中，学生获得的分数来自四部分：代数、几何、数论和组合. 任意两名学生在所有四部分中的分数互不相同. 若一组学生能在至少两部分的分数同时按递增的次序排列，则称该组学生是"好的". 求最小的正整数 N，使得任意 N 名学生中均存在 10 名学生是好的. | 730 | numeric | dapo-math-17k | 12,702 | 1530ad86-8628-4e31-a26b-15c0a9513c93 | f9283d969d2cb4efbf889fec62badbc4 | [] |
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $$n^2+4f(n)=f(f(n))^2\quad(1)$$ for all $n \in \mathbb{Z}.$ Provide the number of distinct functional forms for $f(n)$. | 3 | numeric | dapo-math-17k | 13,522 | 394d1b31-f988-4228-b024-6b4fd3cb7389 | 06ecf33c86fb192422cd336a7af1efee | [] |
13 fractions are formed using each of the numbers $1, 2, \ldots, 26$ exactly once. For example: $\frac{12}{5}, \frac{18}{26}, \ldots$. What is the maximum number of these fractions that can be integers? | 12 | numeric | dapo-math-17k | 1,363 | 5529bc3b-f040-4e62-af90-5d74d93e8b7e | 7d8b78b719b370503283ad3a075b7698 | [] |
A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along... | 871 | numeric | dapo-math-17k | 9,308 | d76d16f4-e056-4321-8c8d-607c9d3764c8 | a83bf5cfb4a13084965533bba3210aa5 | [] |
Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that d... | 50 | numeric | dapo-math-17k | 3,521 | a940ac16-ba3c-4898-a70d-3db6c579bc1d | 3e8bfa482ec4021a00a291d8331ee7e1 | [] |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$. | 243 | numeric | dapo-math-17k | 11,088 | cd2d4a05-0556-46f7-88bf-90dd91a58fa6 | cc33166840346d9ce616720220f78571 | [] |
For positive integers \( n \) and \( k \) with \( n \geq k \), define
\[
a(n, k) = \sum_{j=0}^{k-1} \binom{n}{j} 3^j.
\]
Let
\[
L=\lim_{n \to \infty} \frac{1}{4^n} \sum_{k=1}^n \frac{a(n, k)}{k}.
\]
Choose the correct option to
\[
\lim_{n \to \infty} n \left(4^n L - \sum_{k=1}^n \frac{a(n, k)}{k} \right).
\]
A. ... | B | expression | nemotron-native | null | e1261a19-332b-410f-9170-3eb74920841d | 5f6558b70e779a0ec8424febab4d069a | [] |
Find a particular solution to the PDE $u_{t} - u_{x} = -x$. | \frac{x^2}{2} | expression | nemotron-native | null | ee680230-88b1-48cf-b9b8-d58797e6a5e9 | 09b57a2abaecb19ba8246186149dfa36 | [] |
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, an... | 280 | numeric | dapo-math-17k | 4,861 | 2b326cb5-dc19-41f3-88fa-91b233f36740 | 840bf0f85e6494903fd3f967768bef9d | [] |
Let \( N \) be a positive integer. Two players play a game where the first player writes a list of positive integers not greater than 25, such that their sum is at least 200. The second player wins if they can select some of these numbers so that their sum \( S \) satisfies \( 200-N \le S \le 200+N \). What is the smal... | 11 | numeric | nemotron-native | null | af3a267e-f559-4791-93ce-f05f04e0dbf5 | 0416ff76215aab3f57223e8fe8c0acc5 | [] |
Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. Wha... | \frac{1509}{128} | expression | skywork-or1 | 17,699 | 1aea44cc-a70b-44cd-ad67-763cac06b9a6 | 1266d215ea4eec62244b257092bbc997 | [] |
Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient. | -1 | numeric | dapo-math-17k | 11,294 | 0a081987-cdbe-4cfb-a4e4-f1b971d9dc61 | 3924f746d50b39b778c6532240be1a52 | [] |
将 33\times33 的方格表中每个格染成三种颜色之一，使得每种颜色的格的个数相等. 若相邻两格的颜色不同，则称其公共边为"分隔边". 求分隔边条数的最小值. | 56 | numeric | dapo-math-17k | 12,633 | 6441c578-b4ba-4dec-a918-5b00fd6a8f66 | b54e64f2b16fb74db9e9b804caf4508a | [] |
Find
\[
\int_0^\infty \left(3 - \sqrt{8} \cos(u \ln2)\right) \left| \zeta \left(\frac{1}{2} + iu\right)\right|^2 \frac{du}{u^2 + \frac{1}{4}},
\]
where \(\zeta\) denotes the Riemann zeta function. | \pi \ln 2. | expression | nemotron-native | null | 6c72bea2-72b2-428c-9f3d-31bf20f9a699 | ebe7a858a7321ec498b0e5b6c475b416 | [] |
Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p,q)$ of nonnegative integers satisfying $p + q \le 2016$. Kristoff must then give Pr... | 18 | numeric | dapo-math-17k | 7,412 | 7caa06ed-c6df-4e91-b98c-c8766872bd7f | debdae4d53c194456993e99bf07f6ddd | [] |
Let \(\Lambda=\mathbb Z_2[[T]]\) and for \(n\ge1\) put
\[
\nu_n(T)=\frac{(1+T)^{2^n}-1}{T}\in\Lambda.
\]
Fix \(C_0,C_1\in\mathbb Z_2\) with \(C_0\equiv 2\pmod4\) and \(C_1\equiv0\pmod4\), and define \(P(T)=T^2+C_1T+C_0\).
Let \(G\) be the pro-\(2\) group on generators \(a,b,c\) with relations
\[
[a,b]=a^{-2},\qquad [... | 2^{3n+2} | expression | nemotron-native | null | 1adc61b9-e467-4373-8d54-e0dfeeffe4b9 | 711452be469a6bcd996d6f76f7f2875b | [] |
Let \(R\) be a commutative ring. The extended zero-divisor graph
\(\widetilde{\Gamma}(R)\) is defined as a simple undirected graph whose vertex set is
\(Z(R)^{*}\), the set of all nonzero zero--divisors of \(R\). Two distinct vertices
\(u, v \in Z(R)^{*}\) are adjacent if either \(uv = 0\) or \(u+v \in Z(R)\).
Compu... | 9 | numeric | nemotron-native | null | 1d4a4212-fede-4ec8-b18d-0233016f584c | 04be81ed59b1acd6147ed20db45d4f02 | [] |
Problem 3. On the board, the numbers $2,3,5, \ldots, 2003,2011,2017$ are written, i.e., all prime numbers not exceeding 2020. In one operation, two numbers $a, b$ can be replaced by the largest prime number not exceeding $\sqrt{a^{2}-a b+b^{2}}$. After several operations, only one number remains on the board. What is t... | 2011 | numeric | skywork-or1 | 8,680 | 22eac4ee-ba6e-4c45-9243-253d089ff8b8 | fc0ab58dbc7e4b08bd4087a8791e9f64 | [] |
一块 41\times41 的战场分成 1681 个单位正方形. 某个单位正方形内隐藏着一辆坦克. 一架轰炸机对战场进行轰炸，每次轰炸一个单位正方形. 若某次轰炸坦克所在正方形没有被炸到，则坦克保持在原地不动；若被炸到，则坦克要转移到与其相邻（有公共边）的单位正方形内. 摧毁坦克需要击中其两次，整个过程中轰炸机的飞行员完全不知道坦克的位置及是否被击中. 求可确保摧毁这辆坦克所需要的轰炸次数的最小值. | 2521 | numeric | dapo-math-17k | 12,534 | 1dc4b8a9-99b5-4c4e-a687-ae0f6296e057 | 200fbb601977f67308e462f9ae9b059b | [] |
Let \(G\) be a finite simple graph. A set \(S\subseteq V(G)\) is called an indeque set if the induced subgraph \(G[S]\) is a disjoint union of cliques. The indeque number of \(G\) is the maximum of \(|S|\) over all indeque sets \(S\subseteq V(G)\); denote this maximum by \(\omega_{\alpha}(G)\).
Let \(\mathcal{G}\) be ... | \dfrac{1}{2} | expression | nemotron-native | null | bcfb37c0-9efd-4e12-8ffa-e405d3980bb6 | efa47913afde3b619353709ec4ecbdbd | [] |
In triangle $ABC$ , $AB = 10$ , $BC = 14$ , and $CA = 16$ . Let $D$ be a point in the interior of $\overline{BC}$ . Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$ , respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$ . The maximum possible a... | 150 | numeric | dapo-math-17k | 6,080 | a64f588c-e802-407e-8e26-ad6b6daea5a6 | d555aadd543ea0fb39948658dcd514a3 | [] |
Consider the real integral
\[
I(x) = \int_{0}^{1} e^{-4x t^{2}} \cos\!\bigl(5x t - x t^{3}\bigr)\, dt,
\]
as \(x \to \infty\). Choose the best suitable asymptotic approximation.
CHOICES:\
\[(A)\quad
I(x) \sim \tfrac{1}{4}\sqrt{\tfrac{\pi}{x}}\,e^{-\tfrac{25}{16}x}
\]
\[(B)\quad
I(x) \sim \tfrac{1}{2}\sqrt{\tfrac{\pi}... | D | expression | nemotron-native | null | f67d2f31-eae1-4837-b5f9-14909566ee9f | 9022fd29e35a92402b474d2f89f0c821 | [] |
Let $G$ be the set of polynomials of the form $$ P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, $$where $ c_1,c_2,\dots, c_{n-1} $ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$? | 528 | numeric | dapo-math-17k | 8,961 | 0e6fe1dd-5461-41b1-adee-b9404d34b74a | 2d24295594730ff164baf83dd5416931 | [] |
给定整数 $n \geqslant 2$. 黑板上写着 $n$ 个集合，然后进行如下操作：选取黑板上两个互相不包含的集合 $A 、 B$ ，擦掉它们，然后写上 $A \cap B$ 与 $A \cup B$. 这称为一次操作. 如此操作下去，直到任意两个集合中均有一个包含另一个为止. 对所有的初始状态与操作方式, 当n=5时，求操作次数的最大可能值。 | 10 | numeric | dapo-math-17k | 14,416 | f967edf5-a5d7-4365-bc3e-092b4d491394 | 36b5b09984b18fb9b57824e1e000f06c | [] |
James is standing at the point $(0,1)$ on the coordinate plane and wants to eat a hamburger. For each integer $n \geq 0$, the point $(n, 0)$ has a hamburger with $n$ patties. There is also a wall at $y=2.1$ which James cannot cross. In each move, James can go either up, right, or down 1 unit as long as he does not cros... | \frac{7}{3} | expression | skywork-or1 | 84,035 | dccff835-99f7-4e6c-8a85-5b2e2cfe8f48 | 7d578e7b1ed1c43703d7a72849c578b1 | [] |
$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the line $DA$ intersects $CB$ at $F$. Given that $OE = 13$ and $OF = 14$, let $\cos \angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find $p + q$. | 33 | numeric | dapo-math-17k | 1,715 | 9871fd6c-738d-4547-88bb-3a68a7bbb62c | 3cd1f8f1f228334ea1ab3694d4e55985 | [] |
For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. | 125 | numeric | dapo-math-17k | 6,327 | 763f6104-8dbc-43a9-957f-0ce58f7f3e18 | ce27c4242edddf9da88ec40a4e088ae5 | [] |
将有 2023 个顶点的完全图的每条边染色，使得有公共点的任意两条边颜色不同. 定义每个点所连出的边的颜色为一个颜色集，求不同颜色集个数的最小值. | 3 | numeric | dapo-math-17k | 12,540 | acd8b618-f73d-444a-8b0e-90ba855feeb8 | 5c6a21a7a4dcabf838fe659bda0c45fb | [] |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive intege... | 459 | numeric | dapo-math-17k | 6,355 | cd7ca262-9580-4cdd-9a66-a95886e1dcd4 | 8b26247bfd02c0ea13886ce55537ea0e | [] |
设 A 是一个 11 元实数集，令集合 B=\{uv\ |\ u,v\in A,u\neq v\}. 求 B 的元素个数的最小值. | 17 | numeric | dapo-math-17k | 12,277 | f502cfb8-6d83-4b3e-9fe2-94e4531f3b38 | 836a72c02431adb2d31eba701b03b858 | [] |
Alice, Beth, Carla, Dana, and Eden play a game in which each of them simultaneously closes her eyes and randomly chooses two of the others to point at (one with each hand). A participant loses if she points at someone who points back at her; otherwise, she wins. Find the probability that all five girls win. | \frac{1}{324} | expression | skywork-or1 | 14,778 | 1f010438-61f7-45b8-b28b-36d8c7795aa2 | 1343adafc00ffff393d99ed46f5f5d28 | [] |
In the diagram, what is the value of $y$?
[asy]
draw((0,0)--(18,0),black+linewidth(1));
draw((18,0)--(18,-6),black+linewidth(1));
draw((0,0)--(4,6)--(18,-6),black+linewidth(1));
draw((18,0)--(18,-0.5)--(17.5,-0.5)--(17.5,0)--cycle,black+linewidth(1));
label("$80^{\circ}$",(4.5,5),S);
label("$60^{\circ}$",(1,0),NE);
la... | 50 | numeric | dapo-math-17k | 11,017 | 7557cd99-7e0e-4c33-b718-f2f246ac356e | a9820c091e45e582d2a2db75432ed4f2 | [] |
6.6 All natural numbers from 1 to 50 inclusive are written on the board. Vasya chooses a pair of numbers on the board, the greatest common divisor of which is greater than one, and erases one of them. What is the smallest number of numbers Vasya can leave with such actions? | 8 | numeric | skywork-or1 | 25,870 | 8142d865-c14d-4417-b717-e289904c663b | 615c0c9c0c1dc3ff37817648403410d7 | [] |
求具有下述性质的最小正整数 $k$ ：若将 $1,2, \cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 9 个互不相同的红色的数 $x_{1}, x_{2}, \cdots, x_{9}$ 满足 $x_{1}+x_{2}+\cdots+x_{8}<x_{9}$ ，或者存在 10 个互不相同的蓝色的数 $y_{1}, y_{2}, \cdots, y_{10}$ 满足 $y_{1}+y_{2}+\cdots+y_{9}<y_{10} 。$ | 408 | numeric | dapo-math-17k | 11,686 | 6147e989-c426-4c9a-a0ee-d59c762e2705 | bdc0d5bbd3912be546101909411546da | [] |
A circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two runners start at point $A$ at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. The... | 2128 | numeric | dapo-math-17k | 453 | 3c718355-fba7-4938-a89a-304e679003b7 | d3a51ff554bbed891ce12073251beaec | [] |
设 S=\{1,2,\cdots,98\}. 求最小正整数 n，使得 S 的任一 n 元子集中都可以选出 10 个数，无论怎样将这 10 个数均分成两组，总有一组中存在一个数与另外 4 个数都互质，而另一组中总有一个数与另外 4 个数都不互质. | 50 | numeric | dapo-math-17k | 12,473 | 7d53930a-e4aa-4c76-9abd-c35d109a70d7 | 99f02b7be1801ed18c7042a872b16e10 | [] |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leav... | 5 | numeric | dapo-math-17k | 17,877 | 4a5de1e1-5d1f-4e4e-bdd2-c04b8f8b561b | 2dab56cf8e215563e506ad5bf1699452 | [] |
有 2018 名学生围成一圈. 称一个同学是优秀的，如果他两边站着的人和自己的性别都不同. 求优秀的男生数与优秀的女生数的平方差的最大值. | 509040 | numeric | dapo-math-17k | 12,733 | e2fdc6dd-6c09-4235-97e7-608eddcdb785 | 4902c4aa27dc4e63aa186dcf89eebe75 | [] |
Let \( \{F_n\}_{n \ge 0} \) be the Fibonacci sequence defined by \( F_0 = 0, F_1 = 1 \), and \( F_{n+2} = F_{n+1} + F_n \). Given a real constant \( c > 0 \), evaluate the following limit:
\( L = \lim_{n \to \infty} \sqrt{c F_{2}^2 + \sqrt{c F_{4}^2 + \sqrt{c F_{8}^2 + \dots + \sqrt{c F_{2^n}^2}}}} \)
The indices of ... | \frac{3+\sqrt{5+4c}}{2} | expression | nemotron-native | null | e51d58b9-dd4f-496f-b0b3-73d5cfb630d1 | 008c4edc32d5efbca2ab9c07b870fddc | [] |
13. On a sheet of paper, two regular hexagons are drawn. The smaller one has an area of 18, and a shorter diagonal of the larger hexagon coincides with a longer diagonal of the smaller hexagon. What is the area of the union of the two hexagons? | 29 | numeric | skywork-or1 | 24,404 | 9a7a40ce-4302-43ea-9935-36f340923a02 | 15fce00644fdbf78002d799e28ac9fac | [] |
A square, a regular pentagon, and a regular hexagon are all inscribed in the same circle. The $15$ vertices of these polygons divide the circle into at most $15$ arcs. Let $M$ be the degree measure of the longest of these arcs. Find the minimum possible value for $M$. | 48 | numeric | dapo-math-17k | 8,558 | 2a0a5981-b594-4a7c-9947-9d9f3d2dfc74 | 292f4db2a5ec36d82aad786b5467d2d2 | [] |
Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle ... | 1291 | numeric | dapo-math-17k | 7,760 | 525aa3b4-9525-4850-943a-302f0e520554 | bbde118761c84577a66d3e25ee24150d | [] |
For $0\leq p\leq1/2$,let $X_1,X_2,\ldots$ be independent random variables such that $$X_i=\begin{cases}1&with~ probability~p,\\-1&{with ~probability}~p,\\0&{with ~probability}~1-2p,\end{cases}$$ for all $i\geq1.$ Given a positive integer $n$ and integers $b,a_1,\ldots,a_n$, let $P(b,a_1,\ldots,a_n)$ denote the probab... | 5 | numeric | dapo-math-17k | 13,403 | d1e4a347-8437-4a0e-a367-ce4cf88eb9ac | 44b81f7091da185ba19ec03b30c8c0fa | [] |
The following diagonal is drawn in a regular decagon, creating an octagon and a quadrilateral. What is the measure of $x$?
[asy]
import markers;
for(int i=0; i <=10; ++i) {
draw(dir(360*i/10+90)--dir(360*(i+1)/10+90));
}
pair A = dir(360*0/10+90);
pair F = dir(360*7/10+90);
pair G = dir(360*8/10+90);
pair H = dir(360... | 36 | numeric | dapo-math-17k | 10,080 | ab13af08-e95b-4a42-802d-0aa4d71fa244 | 959ff2927a917e7b62e399866b6ddb8c | [] |
In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter. | 77 | numeric | dapo-math-17k | 2,048 | 7ed82b57-217a-4921-be01-25b183e20f16 | 807e8fb4caf1d6c0346b5dd75903fd20 | [] |
Let \( f(u) = u^3 - \lceil 15u^2 \rceil + \lceil 12u \rceil - 68 \).
For each integer \( k \in \{0,1,\dots,8\} \), define
\[
f_k(u) = \left( \frac{u}{2^k} \right)^3 - \left\lceil 15 \left( \frac{u}{2^k} \right)^2 \right\rceil + \left\lceil 12 \left( \frac{u}{2^k} \right) \right\rceil - 68
\]
Let \( R_k \) be the set of... | 117.132 | numeric | nemotron-native | null | 590877b6-1008-4b9e-b184-ba97a7d8e163 | fc141a591e7505af4fddbcbe4d7e1d26 | [] |
设 a_1,a_2,\cdots,a_{2021} 是整数，满足对 n=1, 2, \cdots, 2016，均有 a_{n+5}+a_n>a_{n+2}+a_{n+3}. 求 a_1, a_2, \cdots, a_{2021} 中最大者与最小者之差的最小可能值. | 85008 | numeric | dapo-math-17k | 12,392 | 422bad05-00ad-4bff-b688-7e53a9c9a286 | 670d9ad72f14d85133df9769ef4ac58f | [] |
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$. | 406 | numeric | dapo-math-17k | 7,503 | 61c005b6-df14-4919-8797-a1ece172f143 | 264aedb5256b5532d0e6ff999d0d88b1 | [] |
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. | 118 | numeric | dapo-math-17k | 2,816 | b4e84f9a-0f81-4e92-b4a6-2eac2eb1afd2 | 91ed1d6f03d6805b3634f868ca14a397 | [] |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | 103 | numeric | dapo-math-17k | 15,128 | 23b86472-139c-4967-93a2-63d0947578bb | 8ea6f2643e7b48d6fa0cd3dbed42fdcb | [] |
What is the maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker? | 32 | numeric | dapo-math-17k | 5,399 | 52bf2f02-9f7f-4451-8273-4076554cd5f8 | 4554a7d6bb886a0a3fa5b3f8c6cc2b9c | [] |
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = \frac{x^2}{16}$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to... | 24 | numeric | dapo-math-17k | 5,328 | c26d2326-f864-4be2-b389-7082880fcbe0 | d73258c01f87ab8b2baa6b18e3271adc | [] |
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.\na) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum ... | 35 | numeric | dapo-math-17k | 13,803 | f2329dd3-f7c1-41a7-bc0e-baf0b90afd9b | 8cd628ab55a5ce38d3e9031471659af7 | [] |
11. In the Cartesian coordinate system, the hyperbola $\Gamma: \frac{x^{2}}{3}-y^{2}=1$. For any point $P$ in the plane that is not on $\Gamma$, let $\Omega_{P}$ be the set of all lines passing through point $P$ and intersecting $\Gamma$ at two points. For any line $l \in \Omega_{P}$, let $M, N$ be the two intersection... | 4 | numeric | skywork-or1 | 96,147 | a2126db2-a616-4c92-8064-3b4a3fdef321 | b9064b2b2e8a8734fcf8b481cb9790c3 | [] |
8. A hotel has 90 vacant rooms, each with a unique key. 100 guests arrive, and keys need to be distributed so that any 90 of them can stay in the 90 rooms, with each person getting one room (assuming there is no limit to the number of keys that can be issued for each room or the number of keys each person can receive).... | 990 | numeric | skywork-or1 | 94,862 | 63a60cbd-79b5-4a5a-9f2c-f49b3c7ccc2e | bad8679bd3b4f48f20711d8cfa991e7f | [] |
Consider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\sqrt{2}$ and $BC = 10\sqrt{3}$. Construct semicircles $P$, $Q$, and $R$ with diameters $AB$, $AC$, and $BC$ respectively, such that the plane of each semicircle is perpendicular to the plane of $ABC$, and all semicircles are on the same side of plane ... | 24 | numeric | dapo-math-17k | 324 | 81afe153-6ff2-42e5-9ea3-cd3bec733912 | 49386a59306750211752d177fec2d99c | [] |
The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a tra... | 3 | numeric | dapo-math-17k | 14,160 | 9f15a624-ac23-4450-a597-02e185d3baf3 | 2262d7fb1ff335ef5ae067fb488e4913 | [] |
Find the least positive integer $n$ for which there exists a set $\{s_1,s_2,\ldots,s_n\}$ consisting of $n$ distinct positive integers such that
$\left(1-\dfrac{1}{s_1}\right)\left(1-\dfrac{1}{s_2}\right)\dots\left(1-\dfrac{1}{s_n}\right)=\dfrac{51}{2010}.$
Please provide the value of $n$. | 39 | numeric | dapo-math-17k | 13,515 | b028bd11-97b9-49da-a648-5e66cad7838b | 8797c6b659cb859c61b39005ba6708e7 | [] |
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all ... | 535 | numeric | dapo-math-17k | 3,826 | 332766f3-d4d4-4c2e-a209-2ad5d6a29573 | 59057534ebd0d6a00982be282e4b62c6 | [] |
9・50 For every permutation $S$ of $(1,2, \cdots, n)$ where $n \geqslant 2$, let $f(S)$ be the minimum value of the absolute differences between every two adjacent elements in $S$. Find the maximum value of $f(S)$. | [\frac{n}{2}] | expression | skywork-or1 | 93,391 | c8b8d800-5d15-418f-bc4a-8e6609eae4d8 | f6099e85b7b50358db2b6678063568c6 | [] |
Let $\Gamma$ be a circle with center $A$, radius $1$, and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$, and diameter $DY$, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ c... | 16 | numeric | dapo-math-17k | 2,249 | fcc987e9-6d5b-44bf-914d-133eeaee0717 | fb462f48e74bab1394c0f433dbd2de9f | [] |
Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$. | 31 | numeric | dapo-math-17k | 8,490 | 4f002b98-519f-43de-a2b0-d911ef9284ff | 71b5836f9e89944875cc44ebedee4544 | [] |
Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$. Let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\... | 12 | numeric | dapo-math-17k | 5,313 | 1823cc0e-d40b-44a8-ba97-e0ad7e221a4a | b8ef46dcf0410e887ba24b71cae56c8e | [] |
3. Among any five nodes of a regular grid paper, there will definitely be two nodes, the midpoint of the segment between which is also a node of the grid paper. What is the minimum number of nodes of a grid made of regular hexagons that need to be taken so that among them, there will definitely be two nodes, the midpoi... | 9 | numeric | skywork-or1 | 98,734 | 3710e2e0-3c5d-4394-878d-3956f279e3c3 | 9664225d873fe54162918df2a726b3ed | [] |
What is the largest order of a non-cyclic torsion subgroup of an elliptic curve over \(\mathbb{Q}(\sqrt{-3})\)? | 18 | numeric | nemotron-native | null | 38aea2bc-31f5-454b-bd1d-3e9e6e3b5e7a | b42470406d8cd6d42a5027cbb87397a4 | [] |
Let \(p = 2017\). Given a positive integer \(n\), an \(n \times n\) matrix \(A\) is formed with each element \(a_{i,j}\) randomly selected, with equal probability, from \(\{0,1,\ldots,p-1\}\). Let \(q_n\) be probability that \(\det A \equiv 1 \pmod{p}\). Let
\[
q = \lim_{n \to \infty} q_n.
\]
If \(d_1, d_2, d_3, \ldot... | 98547790 | numeric | nemotron-native | null | f3ca3c29-394d-4a86-a0c2-3c538255d50f | f5887defbfe348fb3436b993194fff85 | [] |
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2$. | 717 | numeric | dapo-math-17k | 6,302 | 85397002-6cf2-4620-b169-4d3b1159dfd9 | f58fbe1216d1167b58d557ea9d94b366 | [] |
已知集合 $S=\{1,2,3, \cdots, N\}$ 的四个500元子集 $A_{1}, A_{2}$, $A_{3}, A_{4}$ 满足：对任意 $x, y \in S$ ，均存在某个 $i \in\{1,2,3,4\}$ ，使得 $x, y \in A_{i}$ 。求正整数 $N$ 的最大可能值。 | 833 | numeric | dapo-math-17k | 11,820 | 8195313d-4786-478f-8450-8819c6095040 | 5039f00a3b6201dbbe83dd6eaf5341d6 | [] |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $\frac{... | 131 | numeric | dapo-math-17k | 5,829 | b06e78cd-0124-4920-918b-1384171bee5c | 4b4714c3fd9eb8217e25e9bd206448ea | [] |
Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Fi... | 21 | numeric | dapo-math-17k | 14,305 | 0bbb7d1f-5afb-44a7-940c-02c8c4e17743 | 719ca047beddb41780ef886555d54bc1 | [] |
Twelve million added to twelve thousand equals
(A) 12012000
(B) 12120000
(C) 120120000
(D) 12000012000
(E) 12012000000 | 12012000 | numeric | skywork-or1 | 91,702 | 71c2654f-563c-4e69-b32c-4d82eaff83a6 | d29e29b9aa3f0eb70f7c8700ac0a730e | [] |
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 97 | numeric | dapo-math-17k | 10,688 | 5a7a5bdf-7acb-4430-b7dc-a6612c5d9320 | 9f746b8b98d73d6d860a7cd8d03d5df5 | [] |
In right $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 12$, $BC = 35$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\o... | 11 | numeric | dapo-math-17k | 6,075 | 7f25be16-23ec-4bed-875e-735e67892342 | 5f4ebf0cb8d7433a5558ef827e5b9d82 | [] |
4.1. Find the area of the figure defined on the coordinate plane by the system
$$
\left\{\begin{array}{l}
2 \pi\left(x^{2}+y^{2}\right) \leqslant 15 \\
x^{4}-y^{4} \leqslant x y-x^{3} y^{3}
\end{array}\right.
$$ | 3.75 | numeric | skywork-or1 | 22,568 | a7d0d740-04d4-450f-861a-0a9674b9040a | 1c9b1b0a3fd5c7e099ef4916a432db79 | [] |
Bakayev E.v.
In a white $2016 \times 2016$ table, some cells are painted black. We will call a natural number $k$ successful if $k \leq 2016$, and in each of the grid squares with side $k$ located in the table, exactly $k$ cells are painted black. (For example, if all cells are black, then the only successful number i... | 1008 | numeric | skywork-or1 | 25,788 | 3c2b4a2b-3b91-47fe-bf21-2dbd8626be67 | 21745ab43c746ecb84632660fafbbc43 | [] |
For a graph \(G,\) and an integer \(k\ge1\), let \(Z_k(G)\) denote the simplicial complex whose facets are exactly the complements (in the vertex set of \(G\)) of independent sets of size \(k\) in \(G\). Determine the homotopy type of \(Z_9(G) \), where \(G\) is the Cartesian product of the path graphs \(P_2\) and \(... | \vee_{6435}S^{14} | expression | nemotron-native | null | f4104e31-702b-40ad-afbe-4ce9e5a25cb5 | 50d2387ca413cfd6eff4e4351ce93187 | [] |
Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \dots, P_n$ such that $n \ge 2$ and:
- The directed line segments $P_0P_1, P_1P_2, \dots, P_{n-1}P_n$ follow successive coordinate directions: east (for $P_0P_1$), north, west, south, east, etc.
- The lengths of th... | 10053 | numeric | dapo-math-17k | 547 | d70fdcbb-abbc-4d00-b32e-fcdcfd72c97d | 36178714eaca4484d2dd2c1734bb1e3a | [] |
Let \(\mathcal{K}\) be a constant defined by the limit:
\[
\ln \mathcal{K} = \lim_{N \to \infty} \left[ \sum_{k=1}^N k \ln k - \left( \frac{N^2}{2} + \frac{N}{2} + \frac{1}{12} \right) \ln N + \frac{N^2}{4} \right].
\]
Consider the definite integral:
\[
\Psi = \int_{0}^{\infty} \ln(1+x^2) \ln\left( \frac{e^{\pi x} - 1... | 7+\dfrac{2}{3}\ln 2 | expression | nemotron-native | null | 75f3dca0-a0e9-4454-bab4-41582ab6a243 | ec9541cc8a8f1d45e725860f35257526 | [] |
Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square. | 2008 | numeric | dapo-math-17k | 2,844 | 22dc3949-28d2-4546-adad-5c39d8fe03e6 | 87bd94d6dc8c16fd9c87d791dc6e8553 | [] |
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on... | 810 | numeric | dapo-math-17k | 9,649 | e0f3d72c-3902-4624-a261-d5939327b42d | 4a000e2afcdce71c74bbdff7f96f9c87 | [] |
Let G be the set of polynomials of the form P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, where c_1,c_2,\cdots, c_{n-1} are integers and P(z) has n distinct roots of the form a+ib with a and b integers. Calculate the number of polynomials in G. | 528 | numeric | skywork-or1 | 93,897 | 777d6084-b8b0-470b-af8f-f3a699696439 | 4a65f8021738e8e93d5b2724ae33efbf | [] |
There is a row of $100$ cells, each containing a token. For $1$ dollar, it is allowed to interchange two neighboring tokens. Additionally, it is allowed to interchange, with no charge, any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the rever... | 50 | numeric | dapo-math-17k | 515 | 94ed29c3-a14c-41ae-9c0e-ebb8da5c54b1 | ddd24f01cbe20dc1b485568ecfab1bbf | [] |
There are nine buttons labeled \( 1 \) through \( 9 \) that must be pressed according to a certain rule:
The buttons are instructed by a command center in ascending numerical order.
At any moment, the button that was most recently instructed by the command center must be pressed next.
On one particular day, button \... | C | expression | nemotron-native | null | a77849b3-96e2-4bea-a2b2-7adf33d1c36a | ecf407dbda90753abbcb0b869721a8d8 | [] |
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies
\[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]
Determine the maximal value of $kn$. | 6 | numeric | dapo-math-17k | 7,182 | ef39c2cc-6efe-4696-b245-66be567dedb9 | a997ee8b0b07eb26eacf9b8830bd7117 | [] |
We say that a string of digits from $0$ to $9$ is valid if the following conditions hold: First, for $2 \le k \le 4$, no consecutive run of $k$ digits sums to a multiple of $10$. Second, between any two $0$s, there are at least $3$ other digits. Find the last four digits of the number of valid strings of length $2020$. | 9040 | numeric | dapo-math-17k | 7,851 | 4558f45e-9e2c-4023-8e58-eb539841a5e6 | a03289970ff5211b7b0fbf110f30cfb3 | [] |
Example 6 Given that $O$ is the circumcenter of $\triangle ABC$, and $X, Y$ are points on $AC, AB$ respectively, such that $BX$ and $CY$ intersect at $O$, and $\angle BAC = \angle AYX = \angle XYC$. Find the size of $\angle BAC$.
---
The translation maintains the original text's formatting and line breaks. | 50 | numeric | skywork-or1 | 463 | 0eed15bc-06ea-4e4d-bd08-b05fe9454b3a | a99d8575b649babf2cb31939541c9f61 | [] |
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|.$ | 27 | numeric | dapo-math-17k | 6,000 | 8064d181-bc4d-4ac8-bffa-85e5886517ad | 9b1c10933abc156df0ca469d807bbf35 | [] |
In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, the following condition holds:
$$x(c) \cdot y(c) \geq 184,$$
where $x(c)$ is the number of purple cells in the same row as $c$, and $y(c)$ is the number of purple cells in the same colum... | 2024 | numeric | dapo-math-17k | 1,254 | 313330e6-b5c6-49f2-aeb4-24242e450ea3 | e662091cb52c567033d405f79c038445 | [] |
Eight women of different heights are at a party. Each woman decides to only shake hands with women shorter than herself. How many handshakes take place? | 0 | numeric | skywork-or1 | 18,618 | 6899da07-78e3-4b49-9c24-3d0b32ea6882 | 70beb49e4c891411d6fc925c44c939f1 | [] |
给定正整数 $n(n \geqslant 3)$. 对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 $\{1,2, \cdots, n\}$ 的任意三个不同数中都有一个... | 1024 | numeric | dapo-math-17k | 14,413 | a69f13b0-8dbc-4a05-a0a2-febcd240363d | c4097940ad7f69578d8f4df12ed577e5 | [] |
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=2, a_{n+1}=2^{a_{n}}$ 。数列 $\left\{b_{n}\right\}$ 满足 $b_{1}=5, b_{n+1}=5^{b_{n}}$. 若正整数 $m$ 满足 $b_{m}>a_{25}$ ，则 $m$ 的最小值为 $\qquad$ ． | 24 | numeric | dapo-math-17k | 11,467 | 4e0e9331-94da-46e7-b4d4-32d986be7304 | b9079f484d7cd4aa3e1031061349878c | [] |
甲在平面上画了一个正 2021 边形，乙在每个顶点上标一个实数，满足相邻两个顶点的标数至多差 1. 对标数至多差 1 的不相邻的两个顶点，甲在它们之间连一条对角线. 用 d 表示甲连对角线的条数，求 d 的最小可能值. | 2018 | numeric | dapo-math-17k | 12,440 | cf8a821b-039a-4513-9ab0-c0e25e9dc234 | b7eb653c3165f19212ff163e0a992b82 | [] |
From the positive integers, $m, m+1, \dots, m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$. | 15 | numeric | dapo-math-17k | 3,859 | a5e4f4aa-e36a-45b2-82fc-168367d2ceff | 7b5bd06291814717d6ec47f1a6c2ca95 | [] |
Find the largest real number \(T > 1\) with the following property: for any \(10\) distinct real numbers chosen from the interval \([1, T]\), three of them can be selected, in increasing order \(x < y < z\), such that the quadratic equation
\(
x t^2 + y t + z = 0
\)
has no real roots.
Choices
A. \(4^{255}\)
B. \(4... | A | expression | nemotron-native | null | fa156c7d-0ee8-405e-8882-15fbeaa686bc | 126ac498ee049e993b443a70194d736c | [] |
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are elimina... | 3432 | numeric | skywork-or1 | 9,674 | 6ec67f3c-ce0a-41f5-9391-017a6e071914 | 3b37d22cf1b9822645aa594d549fa58c | [] |
The equation
$$ (x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016) $$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting... | 2016 | numeric | dapo-math-17k | 4,321 | 149ddfe5-bf7d-406c-8c5d-f200b26345e3 | 17ba2176373061b5feecc9555fd2bb50 | [] |
What is the correspondence chromatic number of the graph obtained from \(C_{100}\) by replacing each edge with \(1234\) parallel edges? | 2469 | numeric | nemotron-native | null | 16c2dacc-69e7-4635-ac54-a57dcfe401c0 | cae0bb58d8dbdbd290ef57c26e67e248 | [] |
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Nemotron-RL-Math-v2 — reconstructed, de-duplicated, rule-checkable subset
A cleaned derivative of nvidia/Nemotron-RL-Math-v2, prepared so it can be dropped straight into an RLVR pipeline with a rule-based verifier.
1,922 rows. No blank fields, no duplicate problems, no instruction scaffolding, every gold answer parseable by math_verify.
Why this exists
The upstream release is awkward to consume in three specific ways:
- 3,984 of its 7,732 rows ship with an empty
questionand emptyexpected_answer, plus an_hf_question_placeholderrecipe pointing into Skywork-OR1 or DAPO-Math-17k. Loaded naively, over half the dataset is blank. They are reconstructed here following the upstreamfill_placeholders.py. - Every problem is wrapped in one of dozens of "put your answer in
\boxed{}" phrasings (leading and/or trailing), so prompts are non-uniform and the same problem recurs under different wrappers. Wrappers are stripped; a filled\boxed{...}that is genuinely part of a fill-in-the-blank question is preserved. - 47% of upstream golds are prose, equation-shaped (
X = value) or multi-part, which a rule-based verifier marks wrong even when the model is right. Those are removed here.
What was done
| step | rows |
|---|---|
| upstream rows | 7,732 |
| of which reconstructed from placeholders | 3,984 |
dropped: gold not rule-checkable (prose / = / multi-part / long symbolic) |
3,359 |
| dropped: duplicate problem after wrapper stripping | 2,451 |
| kept | 1,922 |
De-duplication is safe: among the 2,451 duplicates there were zero cases of the same problem carrying a different answer.
Columns
| column | meaning |
|---|---|
problem |
problem statement, instruction scaffolding removed |
answer |
gold answer, normalised (delimiters unwrapped, double-escaping fixed) |
gold_type |
numeric (1,410) or expression (512) |
source |
nemotron-native (534) · skywork-or1 (916) · dapo-math-17k (472) |
source_row |
row index in the upstream source, for placeholder-derived rows |
nemotron_uuid |
upstream uuid, so any row can be traced back |
problem_sha1 |
16-byte digest of problem, for dedup across revisions |
alt_answers |
alternative golds seen on a duplicate problem (empty everywhere here) |
151 problems (7.9%) are in Chinese.
Important caveats
- This is deliberately the easy-to-verify slice, not a representative sample of Nemotron.
Upstream declares
verifier_type = math_with_judgeon every row. Filtering to rule-checkable golds removes the problems that actually need that judge. In our measurements the excluded golds are where an LLM judge matters most: prose golds went from 13% to 63% pass under NVIDIA's NeMo-Gym judge, while integer golds were unchanged. - The source mix shifts. Upstream is 53% Nemotron-native by unique problem; this subset is 28%, because 72% of native golds are not rule-checkable. What remains leans toward re-packaged Skywork and DAPO content, which can also be obtained from those datasets directly.
- Still hard, and still long. On a 300-problem sample of upstream Nemotron with Qwen3.5-4B at a 64k budget, the rule-checkable subset scored ~22% with ~55% of responses not finishing inside 64k — the checkable problems are the longest ones.
Provenance and licensing
This is a three-way derivative. Redistribution terms are not uniform, which is why this repo is private:
| upstream | rows here | licence |
|---|---|---|
| nvidia/Nemotron-RL-Math-v2 | all (structure + 534 native problems) | CC-BY-4.0 |
| Skywork/Skywork-OR1-RL-Data | 916 | unstated upstream |
| BytedTsinghua-SIA/DAPO-Math-17k | 472 | Apache-2.0 |
Do not make this public without resolving the Skywork terms. Not affiliated with or endorsed by NVIDIA, Skywork, or the DAPO authors.
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