Datasets:
prompt_id stringlengths 64 64 | question stringlengths 23 5.98k | response stringlengths 439 33.7k | teacher_model stringclasses 1
value | thinking_enabled bool 1
class | generation_attempts int64 1 3 | finish_reason stringclasses 1
value | output_tokens int64 181 12.3k | accepted bool 1
class | stop_reason stringclasses 0
values | rejection_reasons listlengths 0 0 | sampling_seed int64 40.4k 4.29B | teacher_revision stringclasses 1
value | timestamp stringlengths 32 32 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
cab09a021e4eefebe9824b4d5f33a1d2175a31ca9c1c5b22e71daa2833668a86 | Compute
\[\prod_{k = 1}^\infty \left(1 + \frac{1}{k^2 + k}\right).\] | We are asked to compute the infinite product:
$$
\prod_{k = 1}^\infty \left(1 + \frac{1}{k^2 + k}\right).
$$
---
### Step 1: Simplify the term inside the product
We start by simplifying the term inside the product:
$$
1 + \frac{1}{k^2 + k} = \frac{k^2 + k + 1}{k^2 + k}.
$$
So the product becomes:
$$
\prod_{k = 1... | Qwen/Qwen3-4B | false | 1 | stop | 4,052 | true | null | [] | 486,664,709 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.479480+00:00 |
9e765d0a531eddbb9f601800daa8d2e42da09700aea4bc8e8290e872233b2146 | In the figure below, points $A$ and $B$ lie on a circle whose diameter passes through the vertex $V$ of isosceles triangle $\triangle ABC$. If $BA = BC$, the measure of the largest angle formed by drawing line segments connecting points $A$, $B$, and $C$ to point $V$ equals $\tfrac{135}{2}$ degrees, and the measure of ... | We are given a geometric configuration involving an **isosceles triangle** $ \triangle ABC $ with:
- $ BA = BC $
- The **diameter** of a circle passes through vertex $ V $ of the triangle
- $ \angle CBA = \frac{\pi}{12} $ radians
- The **largest angle** formed by connecting points $ A, B, C $ to $ V $ is $ \frac{135}{... | Qwen/Qwen3-4B | false | 1 | stop | 2,623 | true | null | [] | 1,696,414,707 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.335195+00:00 |
0e86a1c95eff2937baa00378dcdf51a99ecff632766f5e4b8c73acfab460dc7b | In the system of congruences
$$\begin{align*}
2x + 3y &\equiv 1 \pmod 5,\\
3x - 2y &\equiv 4 \pmod 7,
\end{align*}$$
find the smallest non-negative integer $n$ for which $x^2 + xy + y^2 + n$ is divisible by $35$. | We are given a system of congruences:
$$
\begin{align*}
2x + 3y &\equiv 1 \pmod{5}, \\
3x - 2y &\equiv 4 \pmod{7},
\end{align*}
$$
and we are to find the **smallest non-negative integer $ n $** such that:
$$
x^2 + xy + y^2 + n \equiv 0 \pmod{35}.
$$
---
### Step 1: Solve the system of congruences
We will solve th... | Qwen/Qwen3-4B | false | 1 | stop | 2,839 | true | null | [] | 2,716,232,281 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:59.904827+00:00 |
df24be5ee2cf8f0ae2e6a89d43712fad2d41acba8fa584027a2d20f94613e4fe | A square pyramid with a base side length of 8 cm and a height of 10 cm is inscribed in a sphere. The pyramid's apex touches the sphere's surface, and its base is tangent to the sphere at the center of the base. What is the radius of the sphere? Express your answer to the nearest whole number. | We are given a **square pyramid** with:
- **Base side length** = 8 cm
- **Height** = 10 cm
- The pyramid is **inscribed in a sphere**, with the **apex touching the sphere** and the **base tangent to the sphere at the center of the base**.
We are to find the **radius of the sphere**, rounded to the nearest whole n... | Qwen/Qwen3-4B | false | 1 | stop | 3,099 | true | null | [] | 1,761,047,145 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:51.841858+00:00 |
a655e8a65a15dc1cb04804bd235c8dae97e8f9613a775fce023943119f863b71 | The function $f(x)$ is defined as follows:
$f(x^2)=xf(x)+\ds\frac{x}{1+x}$ for all $x\in\mathbb{R}$,
while $f(x)$ is an even function.
Determine $f(4)$. | We are given a functional equation and the fact that $ f(x) $ is an **even function**:
---
### Given:
1. $ f(x^2) = x f(x) + \frac{x}{1 + x} $ for all $ x \in \mathbb{R} $.
2. $ f(x) $ is an **even function**, i.e., $ f(-x) = f(x) $ for all $ x \in \mathbb{R} $.
We are to **determine** $ f(4) $.
---
### Step 1: Us... | Qwen/Qwen3-4B | false | 1 | stop | 2,849 | true | null | [] | 3,403,819,902 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.623724+00:00 |
a7eb37b4c8ee1dc717c71006165a05949f4c85f149f5cc603f442db0fb59ee8a | Solve the equation $\frac{x^3-2x^2-x+1}{x^2-2x-1}=\sqrt{\frac{x^2-2x-1}{x+1}}$. | We are given the equation:
$$
\frac{x^3 - 2x^2 - x + 1}{x^2 - 2x - 1} = \sqrt{\frac{x^2 - 2x - 1}{x + 1}}
$$
---
### **Step 1: Domain Restrictions**
We need to ensure that all expressions are defined.
- The denominator $x^2 - 2x - 1 \ne 0$
- The expression under the square root must be non-negative:
$$
\frac{x... | Qwen/Qwen3-4B | false | 1 | stop | 4,635 | true | null | [] | 4,001,629,361 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.117132+00:00 |
f1dd789f8082cba39043e11601cee5a241d63826a51323aab20339fd39318020 | Find the value of $x$ for which the graph of the function $f(x) = \frac{x^2+1}{x^3-2x^2+x+1}$ has a hole. | We are given the function:
$$
f(x) = \frac{x^2 + 1}{x^3 - 2x^2 + x + 1}
$$
We are asked to find the value of $ x $ for which the graph of this function has a **hole**.
---
### Step 1: Understand what a **hole** means
A **hole** in the graph of a rational function occurs when there is a **common factor** in the **n... | Qwen/Qwen3-4B | false | 1 | stop | 714 | true | null | [] | 1,470,096,670 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.832759+00:00 |
3edbddf40e9d47e3c4ac3dfbbb30bf6ed31fe43e7651ef73d1e4b635021fe8cb | Find all points that are 3 units from the parabola $y=x^2$ and 2 units from the point $(1,3)$. What is the sum of the $x$-coordinates of all these points? | We are given two conditions for points $(x, y)$:
1. The point is **3 units from the parabola** $y = x^2$.
2. The point is **2 units from the point** $(1, 3)$.
We are to **find all such points** and then **sum their $x$-coordinates**.
---
### Step 1: Distance from a point to a parabola
The distance from a point $(x... | Qwen/Qwen3-4B | false | 1 | stop | 2,274 | true | null | [] | 2,160,606,547 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:02.958284+00:00 |
5cc0d01e431747e81fd2dd7534e83164f7e5bf53e96a2046fba817e27ce79bdb | Given that
$\ds\sqrt {\f{x^2-3x-2}{x+2}}=x-3$
where $x$ is positive,
determine $\sqrt x$. | We are given the equation:
$$
\sqrt{\frac{x^2 - 3x - 2}{x + 2}} = x - 3
$$
We are to **determine** $\sqrt{x}$, assuming $x$ is **positive**.
---
### Step 1: Square both sides to eliminate the square root
$$
\left( \sqrt{\frac{x^2 - 3x - 2}{x + 2}} \right)^2 = (x - 3)^2
$$
$$
\frac{x^2 - 3x - 2}{x + 2} = x^2 - 6x ... | Qwen/Qwen3-4B | false | 1 | stop | 10,157 | true | null | [] | 3,572,146,232 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.513716+00:00 |
6ed8127f03ff07a5eddc5e572bd548056d0cf68cf0d2c5cfd56f6a8676932798 | In trapezoid $ABCD$, the diagonals intersect at $E$, and $EF$ is the angle bisector from vertex $E$ that intersects side $AD$ at $F$. If $AB = 6$, $CD = 8$, and $EF = 3 \sqrt{3}$, then find the area of triangle $AEF$. | We are given a trapezoid $ABCD$ with the following:
- $AB = 6$
- $CD = 8$
- Diagonals intersect at $E$
- $EF$ is the angle bisector from $E$, intersecting side $AD$ at $F$
- $EF = 3\sqrt{3}$
We are to **find the area of triangle $AEF$**.
---
### Step 1: Use the properties of trapezoid and diagonals
In a trapezoid ... | Qwen/Qwen3-4B | false | 1 | stop | 2,753 | true | null | [] | 1,247,182,954 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.406049+00:00 |
a74a6639355bfe431655522ef98aee96075bcfb9fcef3b32f679ae2999851b98 | Find the largest positive real number $y$ such that
\[\lceil y^3 \rceil - y^2 \lceil y \rceil = 12.\] | We are given the equation:
$$
\lceil y^3 \rceil - y^2 \lceil y \rceil = 12
$$
We are to **find the largest positive real number $ y $** that satisfies this equation.
---
### Step 1: Understand the ceiling function
The **ceiling function** $ \lceil x \rceil $ is the **smallest integer greater than or equal to $ x $... | Qwen/Qwen3-4B | false | 3 | stop | 8,276 | true | null | [] | 320,877,101 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:10.168238+00:00 |
98e5b2d886c9800264194feb1fbb602141ad992986cac0b0a2a6a85a0f8a6162 | Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that
\[g(g(x) + g(y)) = g(x) + g(y) + g(1)\]for all $x,$ $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | We are given a function $ g : \mathbb{N} \to \mathbb{N} $ satisfying the functional equation:
$$
g(g(x) + g(y)) = g(x) + g(y) + g(1) \quad \text{for all } x, y \in \mathbb{N}.
$$
We are to find:
- $ m $: the number of possible values of $ g(1) $,
- $ t $: the sum of all possible values of $ g(1) $,
- and compute $ m... | Qwen/Qwen3-4B | false | 3 | stop | 4,550 | true | null | [] | 4,115,140,755 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:27.747593+00:00 |
431d3d1f1211d739dee2464c7fe92f2196a56c45e4ad9b428b1abc0135b7da22 | In quadrilateral $ABCD$, $AB=6$, $BC=8$, $CD=4$ and $\angle C=90^\circ$. The quadrilateral is inscribed in a circle centered at $O$. If the diagonals $AC$ and $BD$ intersect at $E$, and $\angle AED=90^\circ$, then what is the length of $OE$? | We are given a **cyclic quadrilateral** $ABCD$ with:
- $AB = 6$
- $BC = 8$
- $CD = 4$
- $\angle C = 90^\circ$
- Diagonals $AC$ and $BD$ intersect at point $E$, and $\angle AED = 90^\circ$
- The quadrilateral is inscribed in a circle centered at $O$
We are to find the length of $OE$.
---
### Step 1: Understand the G... | Qwen/Qwen3-4B | false | 1 | stop | 2,525 | true | null | [] | 1,343,601,450 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.275852+00:00 |
98948231653e1b2a141f8355a8b84603475b4887553c3a24cff5642a3ff9417a | What is the value of $b$ if $x^2+bx+1<0$ only when $x\in (0, 2)\cup(4,6)$? | We are given the inequality:
$$
x^2 + bx + 1 < 0
$$
and told that this inequality **only holds** when:
$$
x \in (0, 2) \cup (4, 6)
$$
This tells us that the quadratic expression $ x^2 + bx + 1 $ is **negative** only on the intervals $ (0, 2) $ and $ (4, 6) $, and **non-negative** outside of those intervals.
---
#... | Qwen/Qwen3-4B | false | 1 | stop | 3,135 | true | null | [] | 1,483,133,845 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.844569+00:00 |
41e9074b04f55ba6420b7a5502486ed59b298ffafe2815598a6ec2e5408864f6 | The function $f(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f \left( \frac{x}{3} \right) = \frac{f(x)}{2}$ for $0 \le x \le 1.$
Find $f \left( \frac{2}{7} \right).$
Anothe... | We are given two problems with similar structures. Both involve functions defined on the interval $[0,1]$ with symmetry and scaling properties. We'll analyze and solve each one step by step.
---
### **First Problem: Find $ f\left(\frac{2}{7}\right) $, given:**
1. $ f(0) = 0 $
2. $ f(x) \le f(y) $ if $ 0 \le x < y \l... | Qwen/Qwen3-4B | false | 1 | stop | 2,874 | true | null | [] | 4,010,899,979 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.024942+00:00 |
c331742a7a6513ec8cb34c6bc8eac9f6d1252b8821e90a6bcf90117f46e58fd2 | Consider two functions, $f(x)$ and $g(x)$, such that their graphs intersect at $(1,1)$, $(3,4)$, and $(5,7)$. The graphs of $y=f(x+1)$ and $y=g(x-1)$ intersect at two points. What is the difference between the $y$-coordinates of these two points? | We are given two functions $ f(x) $ and $ g(x) $, whose graphs intersect at the points:
- $ (1, 1) $
- $ (3, 4) $
- $ (5, 7) $
This means:
$$
f(1) = 1,\quad g(1) = 1 \\
f(3) = 4,\quad g(3) = 4 \\
f(5) = 7,\quad g(5) = 7
$$
So, both functions pass through the same three points.
Now, we are told:
- The graphs of $ ... | Qwen/Qwen3-4B | false | 1 | stop | 2,187 | true | null | [] | 1,820,754,744 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.550752+00:00 |
c906888fc94bdbed15c46adc841503362020ad0ab876a6bbc8077b486b3a19f1 | Find the slope of the line that passes through the intersection points of the curves $y = \frac{1}{x^2}$ and $y = x^3$. Express your answer as a common fraction. | We are given two curves:
$$
y = \frac{1}{x^2} \quad \text{and} \quad y = x^3
$$
We are asked to find the **slope of the line that passes through the intersection points** of these two curves.
---
### **Step 1: Find the intersection points**
To find the points of intersection, set the two expressions for $ y $ equa... | Qwen/Qwen3-4B | false | 1 | stop | 1,708 | true | null | [] | 2,766,980,275 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.470224+00:00 |
c6c3f692d17734c039313f2d588ff9ba1f09b9168ff56c7f54b6e855100e2610 | Let $a,$ $b,$ and $c$ be real numbers such that
\[a^4 + b^4 + c^4 - 4abc(a + b + c) = 2.\]Find the minimum value of $a^2 + b^2 + c^2.$ | We are given the equation:
$$
a^4 + b^4 + c^4 - 4abc(a + b + c) = 2
$$
and we are to **find the minimum value of** $ a^2 + b^2 + c^2 $.
---
### Step 1: Use symmetry and assume symmetry in the variables
Let’s try to simplify the problem by assuming that the minimum occurs when $ a = b = c $. This is a common techni... | Qwen/Qwen3-4B | false | 1 | stop | 1,446 | true | null | [] | 4,115,865,990 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:59.951172+00:00 |
6e7c8713d656ed0b87aeac84030af5ea3509023675aba7ebb0b1a61198a075b3 | Solve for $x$: $\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} = 3.$ | We are given the equation:
$$
\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} = 3
$$
We are to **solve for $ x $**.
---
### Step 1: Let’s define a nested expression
Let’s denote:
$$
y = \sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}}
$$
We are told that $ y = 3 $.
So:
$$
\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} ... | Qwen/Qwen3-4B | false | 1 | stop | 2,015 | true | null | [] | 2,550,865,468 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.699408+00:00 |
ddc2ff91ef2849b2164047e3fcf19bbbae18bdab698d73e84c500bdc5340eab5 | A set of six distinct positive integers is such that the sum of any three of them is a multiple of 11. What is the smallest possible sum of all six numbers? | We are given a set of **six distinct positive integers** such that the **sum of any three** of them is a **multiple of 11**. We are to find the **smallest possible sum** of all six numbers.
---
### Step 1: Understanding the Condition
Let the six numbers be $ a_1, a_2, a_3, a_4, a_5, a_6 $, all **distinct positive in... | Qwen/Qwen3-4B | false | 1 | stop | 1,575 | true | null | [] | 4,072,202,130 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.301833+00:00 |
f19aa52ecfb02d7a5092b0b20f255144b439bedbfae293b00a6faf0523226edf | A circle with center at the origin intersects the parabola $y^2 = 2x$ at two points $A$ and $B.$ The tangent to the circle at the point $(\sqrt{2}, 0)$ intersects the line $y = x$ at point $P.$ Find the length of the segment $AP$ multiplied by the length of the segment $BP.$ | We are given:
- A **circle** with center at the **origin** (i.e., $(0, 0)$).
- The **circle intersects the parabola** $ y^2 = 2x $ at two points $ A $ and $ B $.
- The **tangent to the circle at the point** $ (\sqrt{2}, 0) $ **intersects the line** $ y = x $ at point $ P $.
- We are to **find** the product $ AP \cdot ... | Qwen/Qwen3-4B | false | 1 | stop | 2,706 | true | null | [] | 2,985,876,877 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.873940+00:00 |
975f22428b7d38b62ab5ce8566b078ef09892a67ee372329d94a6e08c9956191 | Consider a convex quadrilateral $ABCD$ such that $AB=5$, $BC=4$, and $CD=6$. Find all possible values of the length of $AD$ so that the given quadrilateral can be inscribed in a circle. | We are given a **convex quadrilateral** $ABCD$ with sides:
- $AB = 5$
- $BC = 4$
- $CD = 6$
We are to **find all possible values of $AD$** such that the quadrilateral can be **inscribed in a circle** (i.e., it is **cyclic**).
---
### Step 1: Use the Cyclic Quadrilateral Property
A **cyclic quadrilateral** is one w... | Qwen/Qwen3-4B | false | 1 | stop | 2,924 | true | null | [] | 738,714,274 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.033953+00:00 |
OpenThoughts3 Math — Qwen3-4B Teacher Responses
This dataset contains 21,153 unique math questions with newly generated Qwen3-4B responses, prepared for supervised fine-tuning and teacher–student research. Questions come from the math subset of OpenThoughts3-1.2M. Responses were generated by Qwen3-4B with thinking mode disabled.
Accepted responses passed completion and formatting checks. Mathematical correctness has not been verified, and these responses should not be treated as ground-truth answers or Math-Verify-validated references.
Data preparation
The source revision is 61bcf9d4eb38b30295efc2021227a63cc5bb34c8. Preparation selected rows with domain == math, extracted the human question, and discarded the source assistant completion as a generation target. Original question text is preserved.
Questions were deduplicated using Unicode NFC normalization and collapsed whitespace, preserving case. First-occurrence unique questions were shuffled with seed 42. The source provided 53,106 unique math questions: 30,000 were reserved for future on-policy distillation, and the remaining 23,106 were selected for teacher generation. The reserved pool is excluded from this release.
| Outcome | Questions |
|---|---|
| Selected for teacher generation | 23,106 |
| Accepted and released | 21,153 |
| Failed after exhausting three attempts | 1,953 |
| Reserved separately for future OPD | 30,000 |
Generation finished on October 11, 2026. All selected questions reached a final outcome, with no pending questions. Historical internal filenames included a 200k target; this release contains exactly 21,153 records.
Generation
The teacher revision is 1cfa9a7208912126459214e8b04321603b3df60c. Each question was suffixed with:
Please reason step by step, and put your final answer within \boxed{}.
The official teacher tokenizer applied its chat template with enable_thinking=False. BF16 inference used eight independent vLLM replicas, with one replica per GPU.
| Setting | Value |
|---|---|
| Temperature | 0.7 |
| Top-p | 0.95 |
| Top-k | -1 |
| Min-p | 0.0 |
| Repetition penalty | 1.0 |
| Maximum generated tokens | 12,288 |
| Context capacity | 32,768 |
| Maximum attempts per question | 3 |
| Accepted responses per question | 1 |
| Base seed | 42 |
The format filter uses the authors' rollout validator from Rethinking-OPD. It rejects missing boxed answers and specified forms of repetition. Additional checks reject empty responses, length-truncated generations, and abnormal termination. Failed attempts were retried within the three-attempt limit; replacement questions were not used.
There were 32,193 generation attempts, including 9,087 retries and 11,040 rejected responses. Accepted responses average about 2,847 generated tokens, with a median of 2,429. Questions and accepted raw teacher text are preserved without post-editing.
Files and schema
data/train.parquet is the dataset's single train split. downloads/train.jsonl contains the same records in JSONL format. metadata/ contains generation settings, statistics, validation results, source selection provenance, and file checksums.
| Field | Description |
|---|---|
prompt_id |
Stable identifier for the source question |
question |
Exact original question, without the generation suffix |
response |
Raw accepted teacher response |
teacher_model, teacher_revision |
Teacher identity and pinned revision |
thinking_enabled |
False for every record |
generation_attempts |
Attempt number that produced the accepted response |
finish_reason, stop_reason |
vLLM termination metadata |
output_tokens |
Generated response token count |
accepted, rejection_reasons |
True and an empty list for released records |
sampling_seed |
Seed for the accepted attempt |
timestamp |
Generation attempt timestamp |
from datasets import load_dataset
dataset = load_dataset("zbeeb/OpenThoughts3-Qwen3-4B-Math-21k", split="train")
The JSONL and Parquet exports were validated as equal, with zero duplicate question identifiers. No held-out evaluation split is provided. Deduplication applies within the source-question pool; overlap with external benchmarks has not been checked. Filtering selects for response format and completion, which can bias the released question distribution.
License and attribution
This release uses Apache 2.0, matching the published licenses of OpenThoughts3-1.2M and Qwen3-4B. Credit the OpenThoughts team for source questions, the Qwen team for the teacher model, and the Rethinking-OPD authors for the referenced rollout filtering implementation. Generation and filtering details are provided to support reproducibility; this release does not claim to reproduce the paper's historical 200K dataset.
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