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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
End of preview. Expand in Data Studio

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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