zbeeb/Qwen2.5-3B-GRPO-Staleness-2
Text Generation • 3B • Updated • 290
Datasets:
answers listlengths 1 1 | benchmark stringclasses 1
value | prompt stringlengths 19 2.46k | prompt_tokens int64 69 1.02k | source_id stringlengths 36 36 |
|---|---|---|---|---|
[
"4"
] | dapo_train | Let $C_{1}$ and $C_{2}$ be two externally tangent circles with diameters $[AB]$ and $[BC]$, and centers $D$ and $E$, respectively. Let $F$ be the intersection point of the tangent line from $A$ to $C_{2}$ and the tangent line from $C$ to $C_{1}$ (both tangent lines are on the same side of $AC$). If $|DB| = |BE| = \sqrt... | 202 | 2700c5af-d2d7-4a93-8324-da06387a5210 |
[
"784"
] | dapo_train | How many positive integer multiples of $1001$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$? | 116 | 1a40ae00-90a6-4ef2-b1c3-005698bac072 |
[
"5000"
] | dapo_train | There are $10001$ students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
1. Each... | 202 | fdc101f9-dcee-453b-b8cf-3768536f7cd0 |
[
"120"
] | dapo_train | Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative. | 105 | 5e9e7c89-ab26-4948-9434-85276bfb523e |
[
"31"
] | dapo_train | Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$? | 118 | 2d12800e-9999-4fb7-8391-522a5572dd30 |
[
"7200"
] | dapo_train | 求同时满足下列两个条件的多项式 $f(x)=a x^{3}+b x$ 的个数:
(1) $a, b \in\{1,2, \cdots, 2013\}$;
(2) $f(1), f(2), \cdots, f(2013)$ 中任意两数之差不是 2013 的倍数。 | 149 | 7ae3d1cd-3270-4058-8d79-af5ec62ec7a6 |
[
"11"
] | dapo_train | For each real number $x$, let
$$f(x)=\sum_{n\in S_x}\frac{1}{2^n},$$
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x) ≥ L$ for all $x ∈ [0,1)?$ The original answer is in the format \frac{m}{n}, please provide the value of m + n. | 166 | 47088c4b-781c-4067-be9b-6e5bb6ff139f |
[
"153"
] | dapo_train | Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$ Find $n \times s.$ | 163 | 1213dfd4-98b3-45a6-bf7a-99a26ae11e8f |
[
"2010"
] | dapo_train | 求出所有的正整数 $n$, 使得关于 $x 、 y$的方程 $$ \frac{1}{x}+\frac{1}{y}=\frac{1}{n} $$ 恰有 2011 组满足 $x \leqslant y$ 的正整数解 $(x, y)$。已知答案为p^{m}的形式，给出m 的值。 | 148 | 483ebbfb-540d-421f-94f2-88af2a3fe86c |
[
"24"
] | dapo_train | Given a set $I=\{(x_1,x_2,x_3,x_4) \mid x_i \in \{1,2,\cdots,11\}\}$.
$A \subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4), (y_1,y_2,y_3,y_4) \in A$, there exists $i, j \ (1 \leq i < j \leq 4)$, such that $(x_i - x_j)(y_i - y_j) < 0$.
Find the maximum value of $|A|$. | 193 | 0688eb46-0384-46f9-a6fd-551db179ee94 |
[
"1023"
] | dapo_train | Regular decagon $P_1 P_2 \dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).\] | 176 | 7e0b20f6-3e48-40a0-8681-b17b4e310374 |
[
"4"
] | dapo_train | Let $x,$ $y,$ and $z$ be positive real numbers such that $xy + xz + yz = 1.$ Find the minimum value of $10x^2 + 10y^2 + z^2.$ | 110 | d2ae44d9-2650-4941-a30e-66aaf675b161 |
[
"512"
] | dapo_train | The numbers $a, b, c, d$ are $1, 2, 2, 3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$? | 103 | 59cac655-e4ec-477f-8b2f-c01439ea0611 |
[
"2"
] | dapo_train | Let \( a^3 - a - 1 = 0 \). Find the exact value of the expression
\[
\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.
\] | 114 | 870c674f-7c1a-47cf-a8a1-d72f55af14e2 |
[
"4"
] | dapo_train | For some constants $a$ and $b,$ let \[f(x) = \left\{
\begin{array}{cl}
9 - 2x & \text{if } x \le 3, \\
ax + b & \text{if } x > 3.
\end{array}
\right.\]The function $f$ has the property that $f(f(x)) = x$ for all $x.$ What is $a + b?$ | 155 | 890ac4b9-fe91-4ec4-a2d1-82cf3bb42781 |
[
"761474"
] | dapo_train | For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$. | 169 | 1b3309c1-ed72-4632-98a5-f7bf4cf993d3 |
[
"3"
] | dapo_train | 已知函数 $f(x)$ 在区间 $(0,+\infty)$ 上严 格单调递减，对任意的 $x \in(0,+\infty)$ ，均有
$$
f(x) f\left(f(x)+\frac{2}{x}\right)=\frac{1}{3}.
$$
记 $g(x)=f(x)+4 x^{2}(x \in(0,+\infty))$. 则函数 $g(x)$ 的最小值是 $\qquad$. | 167 | 6396ee6c-c74f-46d6-9252-d5ea12ee6032 |
[
"-15"
] | dapo_train | If $a$ is a root of $x^3-x-1 = 0$, compute the value of $$a^{10 }+ 2a^8 -a^7 - 3a^6 - 3a^5 + 4a^4 + 2a^3 - 4a^4 - 6a - 17.$$ | 135 | a9d95fa7-f0c4-4e21-b85a-ab3b8c7ea410 |
[
"99"
] | dapo_train | In the convex quadrilateral $AEBC$, it is given that $\angle BEA = \angle CAE = 90^{\circ}$, $AB = 15$, $BC = 14$, and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to the line $\overline{AB}$. If the ray $CD$ intersects $\overline{AE}$ at $F$, compute the product $AE \cdot AF$. | 160 | 463c1bd9-a8d9-4172-ac05-69dfe6d386d2 |
[
"0"
] | dapo_train | Let $f(n)$ be a function that fulfills the following properties:
- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.
Find $\displaystyle f(2009^{20... | 199 | 210830fd-5398-4859-9874-4364fe85cdde |
[
"983"
] | dapo_train | 对于集合 $\{x \mid a \leqslant x \leqslant b\}$ ，我们把 $b-a$ 称为它的长度.设集合 $A=\{x \mid a \leqslant x \leqslant$ $a+1981\}, B=\{x \mid b-1014 \leqslant x \leqslant b\}$ ，且 $A, B$ 都是集合 $U=\{x \mid 0 \leqslant x \leqslant 2012\}$的子集，则集合 $A \cap B$ 的长度的最小值是 $\qquad$. | 208 | e8d61bc0-16a4-498c-acc2-0a425bff34e3 |
[
"256"
] | dapo_train | How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.) | 117 | a854c341-0ac3-41e6-bbc8-c5d87226b71b |
[
"10"
] | dapo_train | Five people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *trio* if one of the following conditions is true:
1. $A$ shakes hands with $B$, and $B$ shakes hands with $C$, or
2. $A$ does not shake hands with $B$, and $B$ does not shake hands with $C$.
If ... | 175 | 4cd7cb8a-5ffc-4741-83cd-2e03beaeaa3b |
[
"9"
] | dapo_train | 设实数 x,y,z 满足 x^2+y^2+z^2=1，求(x^2-yz)(y^2-zx)(z^2-xy)的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 130 | b0d1f43a-d8e8-4aa2-8d1f-94ef0d727580 |
[
"258"
] | dapo_train | Let $f(n)$ and $g(n)$ be functions satisfying
$$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$
and
$$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$
for positive integer... | 212 | 6ba70d09-b680-459f-8b84-5f9cc94d4d2c |
[
"91"
] | dapo_train | For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \frac{p}{q}$ for coprime positive integers $p$, $q$, find $p + q$. | 153 | eb833f5b-d1c1-45f3-84d5-c1d86c0d7504 |
[
"13"
] | dapo_train | A sorcerer is concocting a healing potion. In order to do so, he must have one of three roots, and one of five minerals. However, one of the minerals is incompatible with two of the roots. Otherwise, he does not foresee any complications with other combinations of roots and minerals. In how many ways can he brew his po... | 128 | 62b1a340-446e-4c7c-b979-e1f332350af9 |
[
"11"
] | dapo_train | 在直三棱柱 $A_{1} B_{1} C_{1}-A B C$ 中, $\angle B A C=\frac{\pi}{2}, A B=A C=A A_{1}=1$. 已知 $G$与 $E$ 分别为 $A_{1} B_{1}$ 和 $C C_{1}$ 的中点, $D$ 与 $F$ 分别为线段 $A C$ 和 $A B$ 上的动点 (不包括端点). 若 $G D \perp E F$ ，求线段 $D F$ 的长度的取值范围。答案的格式为 \left[\frac{\sqrt{m}}{n}, a\right)，请给出m+n+a的值。 | 222 | 600d6da6-d830-4e65-a41d-478ff9481415 |
[
"7"
] | dapo_train | Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with lengths $5$ and $2\sqrt{6}$, respectively. If $5 + h_1 \leq 2\sqrt{6} + h_2$, find the length of the third side of the triangle. | 129 | fe928764-a14c-452a-8f77-d0443fa9901c |
[
"6"
] | dapo_train | 求所有的正整数 $a$ ，使得对任意正整数 $n \geqslant 5$ ，均有 $\left(2^{n}-n^{2}\right) \mid\left(a^{n}-n^{a}\right)$ 。请提供所有满足条件的正整数 $a$ 的和。 | 127 | 9468bbd3-983d-4697-a7ba-67cd1903f058 |
[
"24"
] | dapo_train | How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) Find the total number of ways. | 124 | ef51d713-f2d0-4314-93b2-c8ff85af9b63 |
[
"36"
] | dapo_train | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $4\times 4$ square array of dots, as in the figure below?
[asy]size(2cm,2cm); for (int i=0; i<4; ++i) { for (int j=0; j<4; ++j) { filldraw(Circle((i, j), .05), black, black); } } [/asy] (Two rectangles are di... | 173 | c1ee4aa4-5694-4396-960a-cc8fd176a04a |
[
"84"
] | dapo_train | Derek's phone number, $336$ - $7624,$ has the property that the three-digit prefix, $336,$ equals the product of the last four digits, $7 \times 6 \times 2 \times 4.$ How many seven-digit phone numbers beginning with $336$ have this property? | 130 | 3f813e50-8bcf-445f-9102-dff3c654a5d3 |
[
"7"
] | dapo_train | The solutions to the equation \((z+6)^8=81\) are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled \(A,B,\) and \(C\). Find the least possible area of \(\triangle ABC\). The original answer is in the form \(\frac{k}{m}\sqrt{n} - \frac{k}{m}\), where k, m, and n are int... | 161 | c58a1ddd-bbb7-400d-9ee5-d8087737c68d |
[
"49"
] | dapo_train | Triangle $DEF$ is shown. What is $\cos{E}$?
[asy]
draw((0,0)--(7,0)--(7,24)--cycle,black+linewidth(1));
draw(rightanglemark((0,0),(7,0),(7,24),30),black+linewidth(1));
label("$E$",(7,24),E);
label("$F$",(0,0),W);
label("$D$",(7,0),E);
label("7",(0,0)--(7,0),S);
label("24",(7,0)--(7,24),E);
[/asy]The answer is in the ... | 231 | ef74de7f-cb2e-4850-b8e8-46035489aaf9 |
[
"54"
] | dapo_train | For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + ... | 194 | 7d53539e-47ae-4d0e-a1cf-9210499cff02 |
[
"30"
] | dapo_train | The vertices of a cube have coordinates $(0,0,0),$ $(0,0,4),$ $(0,4,0),$ $(0,4,4),$ $(4,0,0),$ $(4,0,4),$ $(4,4,0),$ and $(4,4,4).$ A plane cuts the edges of this cube at the points $P = (0,2,0),$ $Q = (1,0,0),$ $R = (1,4,4),$ and two other points. Find the distance between these two points.The answer is in the form ... | 203 | bffbacef-6d50-430b-b44f-52b2dc3d50d1 |
[
"12"
] | dapo_train | Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\pi/81$. Find the remainder when $N$ is divided by $23$. | 116 | 791c8639-35b7-4abe-8af2-1531ea8cd806 |
[
"65"
] | dapo_train | Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is 2, compute the product $(AB)(CD).$ | 133 | 79ad17a3-bfa7-4e7f-8fec-fe67895f8183 |
[
"85"
] | dapo_train | What is the $22^{\text{nd}}$ positive integer $n$ such that $22^n$ ends in a $2$ (when written in base $10$)? | 98 | 0bb0bb1c-25b4-40ee-a8a4-2474e7b53f57 |
[
"48"
] | dapo_train | Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is: | 164 | f594d1f3-dc05-4f0d-ab0b-e00c54f4f72d |
[
"6"
] | dapo_train | How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.) | 131 | 6ca60d5a-cb5f-4446-bfe3-86503f4ebc62 |
[
"49"
] | dapo_train | Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$? | 142 | b2a07802-6e9a-441d-9a79-222299470be0 |
[
"72"
] | dapo_train | Call an ordered triple $(a, b, c)$ of integers feral if $b - a$, $c - a$, and $c - b$ are all prime numbers. Find the number of feral triples where $1 \leq a < b < c \leq 20$. | 119 | b8d323b7-fe5e-4dd3-b284-adfb681890aa |
[
"169"
] | dapo_train | An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1 \leq x < y \leq 26$, there is exactly one passenger that goes from the $x$-th stop to the $y$-th one. If every passenger wa... | 172 | 281c8fb8-fd73-4020-ae50-0cd3d7337e3b |
[
"0"
] | dapo_train | Let \( n \) be a non-negative integer. Define the *decimal digit product* \( D(n) \) inductively as follows:
- If \( n \) has a single decimal digit, then let \( D(n) = n \).
- Otherwise, let \( D(n) = D(m) \), where \( m \) is the product of the decimal digits of \( n \).
Let \( P_k(1) \) be the probability that \( ... | 215 | 2a47dba2-9307-49e0-97c4-babd000626c5 |
[
"9"
] | dapo_train | Compute the largest positive integer $n$ such that $\frac{2007!}{2007^n}$ is an integer. | 86 | 1099d847-f37b-464f-a423-da9c004c9366 |
[
"8"
] | dapo_train | We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$? | 115 | 357043c7-4bff-4b2e-b1a9-f2658989b953 |
[
"135"
] | dapo_train | A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$. | 160 | edf2e3c3-2ec7-4643-88fa-7ec17e5016b8 |
[
"23"
] | dapo_train | At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and give the value of \(k + m\). | 133 | 892236f6-9757-4fcb-ba63-c04f517570ff |
[
"3"
] | dapo_train | In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and... | 191 | 00eab874-29ec-4de4-9454-2894d3271c03 |
[
"88"
] | dapo_train | What is the greatest integer $n$ such that $$n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2014}}?$$ | 116 | eda407a3-8248-4ac6-889f-e2d8c664c7c7 |
[
"6"
] | dapo_train | The digits from 1 to 6 are arranged to form a six-digit multiple of 5. What is the probability that the number is greater than 500,000? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 136 | aece1137-19ac-453d-bb2a-c27604acf734 |
[
"47"
] | dapo_train | A ball is dropped straight down from a height of 16 feet. If it bounces back each time to a height one-half the height from which it last fell, how far will the ball have traveled when it hits the floor for the sixth time, in feet? | 112 | aae56a27-4d73-45d4-a8ef-4097595dbad3 |
[
"83"
] | dapo_train | 若锐角 A,B,C 满足 \sin^2A+\sin^2B+\sin^2C=2，则 \df{1}{\sin^2A\cos^4B}+\df{1}{\sin^2B\cos^4C}+\df{1}{\sin^2C\cos^4A} 的最小值是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 168 | 9c7a1b12-f077-4294-98c6-bbb9c2df6166 |
[
"6"
] | dapo_train | Let $S$ be the smallest set of positive integers such that\n(a) 2 is in $S$,\n(b) $n$ is in $S$ whenever $n^2$ is in $S$, and\n(c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$.\nWhich positive integers are not in $S$? The original answer is in the form of a set, please provide the sum of all positive integers not in $S$.... | 166 | 390297c9-34bc-4d19-8597-9aa4cf1aaf95 |
[
"47"
] | dapo_train | Each day, two out of the three teams in a class are randomly selected to participate in a MATHCOUNTS trial competition. What is the probability that Team A is selected on at least two of the next three days? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provi... | 142 | 72c5db7c-757d-4ea8-b53e-e2f273071006 |
[
"294"
] | dapo_train | Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a cir... | 193 | 7bd10707-b5cc-4365-8eee-6b0687604178 |
[
"7983390"
] | dapo_train | a) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b) How many distinct ways are there of painting the faces of a dodecahedron with 12 different colors? (Colorations are considered distinct if ... | 155 | 0f6a80b1-4ba5-4589-8b67-f22f7973e147 |
[
"285714"
] | dapo_train | Find the smallest positive integer such that when the last digit is moved to the front, the new number is $\frac{3}{2}$ times the original number. | 90 | 35751501-1ed5-4e93-86f2-aa93423597b5 |
[
"8"
] | dapo_train | A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? Express your answer in the form \(\frac{k}{m}\) and find the value of \(... | 132 | 89d42281-001f-40d6-8e3e-ed3392f3d24d |
[
"1092"
] | dapo_train | Let $S_0 = 0$ and let $S_k$ equal $a_1 + 2a_2 + \ldots + ka_k$ for $k \geq 1$. Define $a_i$ to be $1$ if $S_{i-1} < i$ and $-1$ if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_k = 0$? | 164 | 17d906ac-015d-46fb-8099-5492f73ff74d |
[
"88"
] | dapo_train | One right pyramid has a base that is a regular hexagon with side length $1$, and the height of the pyramid is $8$. Two other right pyramids have bases that are regular hexagons with side length $4$, and the heights of those pyramids are both $7$. The three pyramids sit on a plane so that their bases are adjacent to eac... | 225 | b6d5bf91-044e-48e6-9c4a-8b7b9caf15c5 |
[
"6"
] | dapo_train | Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the ro... | 173 | 43b2c9ad-a987-4788-9691-2b133fef41a6 |
[
"93"
] | dapo_train | What is the greatest integer less than 100 for which the greatest common factor of that integer and 18 is 3? | 85 | 68ae1d89-8368-4e9d-b767-04d2f0071491 |
[
"282"
] | dapo_train | 已知首项系数为 1 的五次多项式 $f(x)$ 满足: $f(n)=8 n, n=1,2, \cdots, 5$, 则 $f(x)$ 的一次项系数为 $\qquad$. | 114 | 51c52d2f-8583-4fe8-8a32-fca7d32b74fa |
[
"6"
] | dapo_train | In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.
If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain? | 161 | e6d44b2a-385c-4948-a4ac-1791c5536bc4 |
[
"1"
] | dapo_train | For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1? | 86 | 10490c87-3331-4578-8e41-755ec0d977a6 |
[
"1001"
] | dapo_train | What is the second smallest four-digit number in Pascal's triangle? | 70 | 22f64224-904e-4845-a11a-38fefc85eb97 |
[
"660"
] | dapo_train | Let $S$ be the set of all rational numbers $r$ , $0<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$ , where the digits $a$ , $b$ , and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required? | 148 | 6379fc29-93e6-40a3-86a3-b095eaa9521a |
[
"70"
] | dapo_train | Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$ . Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$ , $[b,c] = 2000$ , and $[c,a] = 2000$ . | 135 | 500c0c17-d01a-49f2-891d-81742ae423ee |
[
"18"
] | dapo_train | How many integers between 123 and 321 inclusive have exactly two digits that are 2? | 80 | cb483fd0-eb46-4a2e-8503-6967bbb51693 |
[
"1370736"
] | dapo_train | Right triangle $XY Z$ has right angle at $Y$ and $XY = 228$, $Y Z = 2004$. Angle $Y$ is trisected, and the angle trisectors intersect $XZ$ at $P$ and $Q$ so that $X$, $P$, $Q$,$Z$ lie on $XZ$ in that order. Find the value of $(PY + Y Z)(QY + XY )$. | 157 | dcf030e7-1c37-4660-9b4c-652dd77a0ebc |
[
"813"
] | dapo_train | A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on the tower... | 191 | 19b84c25-ccca-4f22-8f42-c27ebd2414e1 |
[
"5"
] | dapo_train | A particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse. 假设So=2,Vo=1,The original answer is in the format \frac{m}{n}, please give the value of m+n. | 141 | 03defc4a-a9aa-47ab-a31e-1a72ec152943 |
[
"2005"
] | dapo_train | The polynomial $R(x)$ is the remainder when $x^{2007}$ is divided by $x^2 - 5x + 6$. Express $R(0)$ in the form $ab(a^c - b^c)$. Find the value of $a + c - b$. | 121 | d406fa0f-2e48-472f-8e6a-a2fed2be8520 |
[
"2"
] | dapo_train | 已知实数 $x, y, z$ 满足: $x \geq y \geq z, x+y+z=1, x^{2}+y^{2}+z^{2}=3$. 求实数 $x$ 的取值范围。请提供区间端点的整数部分的和。 | 131 | dd21aea5-7b34-4e7f-bf0e-f3c8d22d6c6d |
[
"18"
] | dapo_train | Suppose the roots of the polynomial $x^2 - mx + n$ are positive prime integers (not necessarily distinct). Given that $m < 20,$ how many possible values of $n$ are there? | 102 | e6a080e2-5be5-4b63-b1c3-46b80db9b711 |
[
"54"
] | dapo_train | The vertices of Durer's favorite regular decagon in clockwise order are labeled as $D_1, D_2, D_3, \ldots, D_{10}$. Determine the angle between the diagonals $D_1D_3$ and $D_2D_5$. | 120 | c4ac1be6-8539-4ba6-bd02-8cae25f5542c |
[
"129"
] | dapo_train | In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30.$ Find the sum of the four terms. | 104 | d1e72bf7-64ed-4f12-ad2d-ef9bd79c87b2 |
[
"1991"
] | dapo_train | Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations
$x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$.
Find the maximum possible value of $x + 1000y$. | 125 | 41e70f41-421f-44a7-a9f1-5a2dd72505cf |
[
"831"
] | dapo_train | Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets $0$ points. The total points are accumula... | 194 | f6f84e44-3281-46ff-9f21-6d536230bbc6 |
[
"19"
] | dapo_train | A game of solitaire is played as follows. After each play, according to the outcome, the player receives either a or b points (a and b are positive integers with a greater than b), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is ... | 141 | a9fb4363-dc0f-42d4-b896-9a9f3afd0671 |
[
"7"
] | dapo_train | Two circles have a radius of $9$, and one circle has a radius of $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown in the figure. The sine of the base angle of the triangle is $\frac{m}{n}$, where $m$ and $n$ are relati... | 148 | 4d4ea0ff-7e43-4206-bd55-f6f8ec71cb1d |
[
"49"
] | dapo_train | For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of such integers. | 105 | fbefdeb1-f237-4475-b83d-9a7d1f1203d0 |
[
"66"
] | dapo_train | Let $S$ be the set of points in the Cartesian plane that satisfy
$\Big|\big||x|-2\big|-1\Big|+\Big|\big||y|-2\big|-1\Big|=1.$
If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by... | 169 | 0f6b41c8-7d16-4e25-949e-b9c886154709 |
[
"75"
] | dapo_train | In English class, you have discovered a mysterious phenomenon: if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes ... | 219 | 4c2e4faa-0e94-498a-86d5-17e11462d281 |
[
"33725"
] | dapo_train | Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$. | 142 | 98e1f1fb-b419-4c52-a636-4c807431c653 |
[
"972"
] | dapo_train | What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\] | 105 | b3292dc1-f97c-48a6-a34c-c365279e849d |
[
"30"
] | dapo_train | We take $100$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{100}$. Determine the last two digits of the number $a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}$. | 126 | 8bf4520d-9032-4808-81a6-5df2d58cee77 |
[
"137"
] | dapo_train | 在 $1,2,3, \cdots, 10$ 中随机选出一个数 $a$ ，在 $-1,-2,-3, \cdots,-10$ 中随机选出一个数 $b$ ，则 $a^{2}+b$ 被 3 整除的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 159 | 6b04e358-e5fe-4c95-bd3e-99d5e35e0053 |
[
"89"
] | dapo_train | Jon wrote the $n$ smallest perfect squares on one sheet of paper, and the $n$ smallest triangular numbers on another sheet (note that $0$ is both square and triangular). Jon notices that there are the same number of triangular numbers on the first paper as there are squares on the second paper. However, if $n$ had been... | 179 | fe156fdd-5af6-4198-9b60-93d5d89c0d7a |
[
"35"
] | dapo_train | 实数 $a, b$ 满足 $\left(a^{2}+4\right)\left(b^{2}+1\right)=5(2 a b-1)$, 如果 $b\left(a+\frac{1}{a}\right)$ 的值为 $x$，求 $10x$ 的值。 | 129 | 99d11ca8-7d48-446e-be87-947ff56fb8fa |
[
"14"
] | dapo_train | For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b, c,$ and $d$, each to some positive power. What is the value of $N$? | 127 | 67a6e9d8-9c81-4889-9d7c-440bc2129acf |
[
"801"
] | dapo_train | What is the remainder when $7^{8^9}$ is divided by $1000?$ | 78 | 5b8fd727-b176-4e42-9b3d-f43f4510941c |
[
"7"
] | dapo_train | Let $x, y$ be positive integers such that:
\[
x^4 = (x-1)(y^3 - 23) - 1
\]
Find the maximum possible value of $x + y$. | 105 | 27dd99c0-c8b6-4595-9d99-06f9abe8d707 |
[
"107"
] | dapo_train | In the figure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, find the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice | 167 | 59deb856-f843-4f41-a527-f0b6d9ce7654 |
[
"13"
] | dapo_train | Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size he... | 173 | 336426ea-3e42-45c2-aba8-0c1b2a3d432d |
[
"2"
] | dapo_train | 在平面直角坐标系中，将圆心在 $y$ 轴上，且与双曲线 $\Gamma: x^2-y^2=1$ 的两支各恰有一个公共点的圆称为 "好圆"。已知两个好圆外切于点 $P(2,0)$ ，存在常数 $\lambda=2$ 及 $x$ 轴上的定点 $A(0,0)$ ，满足若两个好圆外切于点 $P$ ，则它们的圆心距 $d=\lambda|P A|$ 。求 $\lambda$ 和 $A$ 的坐标绝对值之和。 | 195 | 4adf92ad-f9f0-49d0-968c-575988345003 |
[
"6"
] | dapo_train | Square \(ABCD\) has area \(36,\) and \(\overline{AB}\) is parallel to the \(x\)-axis. Vertices \(A,\) \(B\), and \(C\) are on the graphs of \(y = \log_{a}x,\) \(y = 2\log_{a}x,\) and \(y = 3\log_{a}x,\) respectively. The original answer is in the form \(\sqrt[k]{3}\). Please determine the value of \(k\). | 171 | 0209e8b8-c475-4375-8a38-afd1d52e9818 |
End of preview. Expand in Data Studio
Staleness GRPO DAPO Math 17k
The exact 17,005-row training dataset shared by the staleness-cap-2 Qwen2.5-Math-1.5B, Qwen2.5-3B, and Qwen2.5-Math-7B checkpoints, and the staleness-cap-4 Qwen2.5-Math-1.5B checkpoint. All four training manifests record the same SHA-256 for the training file.
Source and processing
Derived from the all configuration of open-r1/DAPO-Math-17k-Processed, itself processed from BytedTsinghua-SIA/DAPO-Math-17k. Source revision: 31dd309567e3da778038cc87d868b6097a3ccf68.
Processing started from 17,398 rows, removed 213 canonical duplicate extras, excluded all five conflicting-answer groups (10 source rows; the duplicate-extras count already includes one extra per conflicting group), removed 112 automatically matched evaluation overlaps and 55 manually audited variants, and removed 8 prompts exceeding 1,024 tokens with the training prompt/template. Rows were shuffled with seed 42. The resulting file is copied byte-for-byte from the completed training runs.
Overlap filtering used the retained MATH-500, AMC23, AIME24–26, Minerva Math and text/final-answer OlympiadBench evaluations. These checks do not establish absence of pretraining contamination or exhaustive near-duplicate removal. This repository contains training data; it does not bundle the benchmark test sets.
Load
from datasets import load_dataset
data = load_dataset("zbeeb/Staleness-GRPO-DAPO-Math-17k", split="train")
Each record preserves the original training fields, including problem text, accepted final answers, source identifier, benchmark label, and prompt-token count. These are math problems with terminal-answer targets, not generated reasoning traces. Prompt-token counts correspond to the training tokenizer and prompt template, not arbitrary tokenizers.
Integrity
- Rows: 17,005
data/train.jsonlSHA-256:285a7b92a3b5a8efee80bda7506764f7ffd42a195303bc9cb77c399927a2b2c3- Source and filtering provenance: dataset-provenance.json
Attribution and license
Credit to the DAPO authors and the Open R1 processing work. The original DAPO dataset declares Apache-2.0; this redistribution preserves that license and includes its text in LICENSE. The processed upstream card does not add a separate license declaration. See DAPO: An Open-Source LLM Reinforcement Learning System at Scale for the source work.
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