Is Human-Written Data Enough? The Challenge of Teaching Reasoning to LLMs Without RL or Distillation
Paper • 2507.09850 • Published • 4
problem stringclasses 50
values | solution stringlengths 2.93k 21.3k | expected_answer stringlengths 1 699 | solution_type stringclasses 5
values |
|---|---|---|---|
A mailman is delivering mail to 10 houses on one side of the street. No two adjacent houses both get mail, and at most three houses in a row don't get any mail. How many ways can this occur? | We have 10 houses in a row on one side of the street. The mailman can either deliver mail to a house (let's denote this as "1") or not deliver mail to it (let's denote this as "0"). So, for each house, there are two possibilities: 1 or 0. However, there are two constraints:
\begin{itemize}
\item no two adjacent hou... | 69 | human_stage1 |
A sequence $\{a_n\}$ is defined by $a_1=1$, and for $n \geq 2$, $a_n$ is the $x$-coordinate of the midpoint of the points of intersection between the line $y=a_{n-1}x$ and the curve $y=|x^2-1|$. Find $\lim_{n\to\infty} a_n$. | I am given a sequence $a_n$ defined over the reals. I know that $a_1 = 1$ and $a_n$ is defined by $a_{n-1}$
as the midpoint between the two points created by intersecting $y_1 = a_{n-1}x$ with $y_2 = |x^2 -1|$.
Let's find a formula for $a_n$ in terms of $a_{n-1}$.
Let $a_{n-1} = \alpha$, assume $\alpha > 0$, then fo... | \lim_{n\to\infty} a_n = \sqrt{\frac{4}{3}} | human_stage1 |
By choosing different real values for \( a \) in the system of equations:
\[ x^2 - y^2 = 0 \]
\[ (x - a)^2 + y^2 = 1 \]
the number of real-valued solutions can be:
a) 0, 1, 2, 3, 4, or 5
b) 0, 1, 2, or 4
c) 0, 2, 3, or 4
d) 0, 2, or 4
e) 2 or 4 | Hmm. Let me try to understand the problem better. It seems like we are given a few equations and asked for the number of real-valued solutions. Real-valued is interesting. That just means that the value can be any real number. We only care about the number of solutions, not the value of them. From what I remember, a so... | \text{(c) 0, 2, 3, or 4} | human_stage1 |
Determine all functions \( f: \mathbb{Q} \to \mathbb{Q} \) such that
\[ f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2} \]
for all \( x, y \in \mathbb{Q} \). \\ | For this problem, I need to find all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that:
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x, y \in \mathbb{Q}$. I'm not really sure how to systematically start solving this but my first instinct is to try some simple substitutions (i.e, plug in some "i... | f(x) = x^2 + \frac 14 | human_stage1 |
Find a particular solution to the differential equation \( y'' - y' - 2y = x \cos(2x) - 3e^{-x} \) using the method of undetermined coefficients. Specifically, focus on finding the particular solution for the term \( x \cos(2x) \). | So let's start by writing out our equation again:
$$y'' - y' - 2y = x\cos(x) - 3e^{-x}.$$
We can find the homogenous/complementary solution first as
$$y'' - y' - 2y = 0$$
This has a characteristic equation of
$$r^2 - r - 2 = 0 \implies (r-2)(r+1) = 0\implies r_1 = 2, r_2 = -1$$
Now, I want to choose an ansatz that is... | y_{p1}(x) = \left(-\frac{3}{20}x - \frac{1}{25}\right)\cos(2x) + \left(-\frac{1}{20}x + \frac{19}{200}\right)\sin(2x) | human_stage1 |
Find all $a \in \mathbb{R}$ such that there exists a function $f: \mathbb{R} \to \mathbb{R}$ satisfying equation
\[ f(x + f(y)) = f(x) + a \lfloor y \rfloor \]
for all $x, y \in \mathbb{R}$. | This is a functional equation where we are requested to find a function satisfying the above relationship. More precisely, our task is to find all real numbers $a$ such that the functional equation has a solution.
The simplest functions are constants, so let us try $f(x) = c$ where $c$ is some real number. In this case... | a = 0 \text{ or } a = 1 | human_stage1 |
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x)-y^{2})=f(x)^{2}-2f(x)y^{2}+f(f(y)).\] | The question is to find all functions $f$ which is defined on the range of real numbers that satisfies the given relationship.
Let's closely analyze the given function first. The function $f$ is a function mapped from the set of real numbers to real numbers. This does not give us any further hints on the question as th... | f(x)=0, \text{ and } f(x) = x^2 | human_stage1 |
Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all positive integers \( n \), there exists a unique positive integer \( k \) satisfying \( f^k(n) \leq n + k + 1 \). | So, I want to try to first understand how this function works. First, I note that $f^k (n)$ in this case generally means the $k$th iterate of the function. This is because for functional equations, this is typically the norm.
I try to explore further how the function works, so I try playing around with different value... | f(n) = n + 2 | human_stage1 |
Find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(xy) = f(x)f(y) \). | Let's look at what could possibly be the solution set. $f(x) = x$ obviously works, as $xy = x*y$ by definition.
\bigskip
However, inspecting further yields $f(x) = x^2$ works too, because $x^2y^2 = (xy)^2$. In fact, this should hold for any power $n$ ($x^ny^n = (xy)^n$)... so, we actually have $f(x) = x^n$ works for A... | \parbox{4in}{$f(x) = 1$, $f(x) = 0$, and every function with $f(0) = 0$, $f(-1) = \pm 1$, and letting $f$'s values on primes be arbitrary, satisfies $f((-1)^ap_1^{\alpha_1}\ldots p_k^{\alpha_k}) = f(-1)^af(p_1)^{\alpha_1}\ldots f(p_k)^{\alpha_k}$ for all other integers.} | human_stage1 |
Find all integers \( m \) and prime numbers \( p \) such that \(\frac{m^3 - pm + 1}{m^2 + pm + 2}\) is an integer. | The problem is asking for all integers $m$ and prime numbers $p$ such that $\frac{m^3 - pm + 1}{m^2 + pm + 2}$ is an integer. For a fraction to be an integer, the denominator must divide the numerator evenly. So the key insight is that $m^2 + pm + 2$ must divide $m^3 - pm + 1$ with zero remainder. This problem actuall... | (m, p) = (1, 2), (-1, 2), (-3, 2) | human_stage1 |
Find all integral solutions to the equation $x^{2}+2y^{2}=z^{2}$. | Okay, so the questions asks to find all integer solutions to the equation $x^2+2y^2=z^2$.
This type of problems seems very similar to Pythagorean triples which satisfy $x^2+y^2=z^2$ for positive integer $x,y,z$. The main difference in this case is that the coefficient of $y^2$ is now $2$ instead of $1$.
Something I k... | (x, y, z) = (k\,\lvert 2a^2 - b^2 \rvert, 2kab, k\,(2a^2 + b^2) \text{ where } b \text{ is odd and } \gcd(a, b) = 1 | human_stage1 |
Find all pairs \((x, y, z)\) of non-negative integers such that \(4^x + 4^y - 1 = z^2\). | In this problem we are asked to find non-negative integers $(x,y,z)$ which satisfy the given equation,
\[ 4^x + 4^y - 1 = z^2. \]
Approaching the question algebraically, we can rearrange the equation to be $$z^2+1=4^x+4^y$$. Then, we can factorize $$z^2+1=(z-i)(z+i)$$. It is clear that for positive integers $x$ and $y... | (0,0,1), (0,1,2), (1,0,2) | human_stage1 |
Find every polynomial \( P(x) \) such that
\[ P(x^k) - P(kx) = x^k P(x) \]
for every real number \( x \) and a given natural number \( k \geq 1 \). | This problem looks rather strange. Maybe, we can try to plug in some values to get some intuition as to what is going on. These polynomial problems will usually fall by some simple or clever substitutions leading us to some properties of the polynomial (usually, some kind of factor or root), and if we factorize, then w... | \text{$P(x) = 0$ works for all k, $P(x) = a(x^2-4)$ for any $a$ when $k=2$.} | human_stage1 |
Find integers \(a\) and \(b\) such that \(a^2 + 2b^2 = 5ab + a\). | At first glance, this looks like a very difficult diophantine. The first thing to notice is that there are terms of mixed degree, meaning there is a term of degree $1$ ($a$) and multiple terms of degree $2$ ($a^2$, $2b^2$).
This means that standard methods we are used to when solving Diophantines, e.g., Simon's Favori... | (a, b) = (-8, -2), (-1, -2), (0, 0), (1, 0) | human_stage1 |
Find the largest real number $k$ for which the inequality
$$(x+y+z)^2-81 \geq k(xy+yz+zx-27)$$
holds true for all $x, y, z > 0$ satisfying
$$xy + yz + zx = xyz. | Let us repeat the problem statement and try to make sure we understand it. So we want to find the largest real number $k$ which satisfies this inequality $$(x+y+z)^2-81 \geq k(xy+yz+zx-27)$$
which should hold true for all positive $x,y,z$ that satisfy some constraint $xy + yz + zx = xyz$. So plenty of $k$ can trivially... | k = \frac{15}{4} | human_stage1 |
Find the locus of points in space which are equidistant from three planes intersecting in pairs and not passing through the same straight line and perpendicular to a plane $\pi $. | To better understand the problem, let's restate it: We are tasked with finding the locus of points in space that are equidistant from three planes. These planes intersect in pairs but do not all pass through the same straight line, and they are perpendicular to a reference plane $\pi$. This setup suggests a specific sp... | \begin{gathered}
\text{The locus consists of four vertical lines in 3D space:} \\[0.5em]
\begin{aligned}
& \text{The incenter line: } \\
& \left\{ \left( \dfrac{uA_x+vB_x+wC_x}{u+v+w}, \dfrac{uA_y+vB_y+wC_y}{u+v+w}, z \right) : z \in \mathbb{R} \right\} \\[1em]
& \text{The three excenter lines: } \\
& \left\{ \lef... | human_stage1 |
Find the maximum and minimum value of \(x^2 + y^2\) given the equation \((x^2 - y^2 + 1)^2 + 4x^2y^2 - x^2 - y^2 = 0\). | Let us restate the problem: we are required to find the maximum and the minimum value of the expression $x^2 + y^2$ where $x$ and $y$ are real numbers satisfying a constraint given by the equation:
\[(x^2 - y^2 + 1)^2 + 4x^2y^2 - x^2 - y^2 = 0\]
In order to solve the problem we can try to apply some optimization method... | \text{Maximum:}\dfrac{3 + \sqrt{5}}{2}, \text{Minimum: }\dfrac{3 - \sqrt{5}}{2} | human_stage1 |
Find the minimum and maximum values of the function \( y = \sin^5 x + \sqrt{3} \cos x \). | Let's start:
The maximum and minimum values of \( \sin x\) are \(1\) and \(-1\).
The maximum and minimum values of \( \cos x\) are also 1 and -1. Thus, \( \sin^5 x \) will also just range between \( 1 \) and \( -1 \) because raising something to an odd power does not change the sign. And \( \sqrt{3} \cos x \) will be b... | \text{$-\sqrt{3}$,$\sqrt{3}$} | human_stage1 |
Find the sequence \(a_n\) such that \(a_0 = 1\) and \(a_{n+1} = \left(1 + \frac{1}{2^{2^n}}\right)a_n + 2^{-2^n}\) for all \(n = 0, 1, 2, \ldots\). | I am given some initial conditions about an integer sequence and a rule that tells me how to compute the next element of the sequence given the previous element of the sequence. Such formulas explaining how to compute an element of a sequence based on some previously computed elements of the sequence are called recurre... | a_n = 3 - \frac{1}{2^{2^n - 2}} | human_stage1 |
For what values of \( k \) does the inequality
\[ a^4 + b^4 + c^4 + d^4 \geq \frac{(a^2 + b^2 + c^2 + d^2)^2}{4} + \]
\[ + k(a^2 + b^2 + c^2 + d^2) \left[ (a-b)^2 + (a-c)^2 + (a-d)^2 + (b-c)^2 + (b-d)^2 + (c-d)^2 \right] \]
hold for all \( a, b, c, d \geq 0 \)? | First we can restate the problem. We want to consider possible values of $k$ which satisfy a long inequality for all non negative $a,b,c,d \geq 0$. The long inequality is: \[ a^4 + b^4 + c^4 + d^4 \geq \frac{(a^2 + b^2 + c^2 + d^2)^2}{4} + \]
\[ + k(a^2 + b^2 + c^2 + d^2) \left[ (a-b)^2 + (a-c)^2 + (a-d)^2 + (b-c)^2 + ... | k \le \frac{1}{12} | human_stage1 |
Given \(a > b > c\), \(a + b + c = 1\), and \(a^2 + b^2 + c^2 = 1\), find the range of values for \(a + b\). | Ok, so it seems like we are given equations and constraints and we must find a range of possible values for $a + b$. We are given the equations $a + b + c = 1$, $a^2 + b^2 + c^2 = 1$, and the constraint $a > b > c$. I wonder if we can just simply find the upper and lower bounds for $a + b$. This approach seems promisin... | \text{$a+b \in (1,\frac{4}{3})$} | human_stage1 |
End of preview. Expand in Data Studio
nvidia/Nemotron-Math-HumanReasoning（複製）
データの内容
OpenMathReasoning の数学問題に、オリンピック数学に深く通じた学生が書いた解答を付けたデータ。解答には、人手で書いた版と QwQ-32B-Preview が生成した版がある。1 行が 1 組の問題と解答で、問題文は problem、正解は expected_answer、解答の出どころは solution_type で分かる。想定される用途は言語モデルの訓練で、ライセンスは CC BY-NC 4.0 のため非商用に限られる。
データの概要
| 項目 | 値 |
|---|---|
| 行数 | 250（1 構成・1 分割の合計） |
| 列数 | 4 |
| 容量 | 2.40 MB（ファイル 3 本） |
| ファイルの形式 | jsonl 1 |
| 公開 | public |
構成と分割
| 構成（config） | 分割（split）ごとの行数 | 列数 |
|---|---|---|
default |
train 250 |
4 |
列（構成 default・分割 train）
| 列 | 型 | 先頭の行の値 |
|---|---|---|
problem |
string |
A mailman is delivering mail to 10 houses on one… |
solution |
string |
We have 10 houses in a row on one side of the st… |
expected_answer |
string |
69 |
solution_type |
string |
human_stage1 |
行数と列は HF のデータセットビューアの集計（元のデータセットの現在の版で数えたもの）。全データセットの一覧は HYUKI5/dataset-catalog にある。
複製について
nvidia/Nemotron-Math-HumanReasoning をリビジョン 188ba91e9c7b1cda9d1b1bca43ff23f4f94cf00a の時点でそのまま複製したもの。中身は変えていない。判断モデル Laya（convaiinnovations/laya）の微調整に使う候補として、調査メモ（Laya 学習用データセット調査）に載せたデータを集めている。
出典
- 元のデータセット: https://huggingface.co/datasets/nvidia/Nemotron-Math-HumanReasoning
- 複製したリビジョン:
188ba91e9c7b1cda9d1b1bca43ff23f4f94cf00a（2025-07-16 00:03:01+00:00） - 元のライセンス表記:
cc-by-nc-4.0
論文
アノテーションの方法
データセットの作成日:
06/01/2025
微調整に使うときの注意
- 調査メモの台帳での区分: C（NC・ND・研究専用・評価専用など、商用の利用を妨げうる記載がある）。区分は記載の文言で機械的に付けたもので、法的な判定ではない。学習に入れる前に元のカードとライセンスの原文で確かめる。
- Laya の入力は 2,048 token まで（ModernBERT-large / mmBERT）。長い本文は先頭から黙って切られるので、切れる行は事前に除くか、要約・切り出しをしてから入れる。
- 分割は記事・文書単位で行い、同じ本文から作った別の問いを学習側と較正側に分けない。温度と閾値は学習に使っていない分割で別々に当てる。
調査メモの記述
| メモ | 節 | ライセンス（記載） | 確認 | 注記 |
|---|---|---|---|---|
nvidia_nemotron.md |
付録. 取得した nvidia 組織のデータセット全件の簡易一覧 (L354) | cc-by-nc-4.0 | 確認済 | 確認印なし(各ファイルの凡例により確認済扱い) ／ 原文注記: × 数学。pass_rate 等は難易度ラベルだが金融と無関係 |
元のカード（日本語訳）
以下は元のデータセットのカードの本文を日本語に訳したもの（機械翻訳）。訳に疑問があるときは、末尾の原文を確かめる。
この論文に付随するデータセット: The Challenge of Teaching Reasoning to LLMs Without RL or Distillation。
すべての実験は NeMo-Skills を使って行われた。
データセットの説明:
Nemotron-Math-HumanReasoning は、人間が書いた数学問題の解答を集めた小規模なデータセットであり、DeepSeek-R1 などのモデルの、拡張された推論スタイルを模倣するように設計されている。これらの解答は、オリンピアード数学に深い経験を持つ学生が書いたものである。私たちは解答の複数のバージョンを提供し、人間が書いた解答で訓練した場合と、QwQ-32B-Preview が生成した解答で訓練した場合の結果を比較する。このリリースが、効果的な「推論スタイル」の解答に寄与する重要な要素の研究を支えることを期待している。 このデータセットは非商用利用に限られる。
私たちは、2 種類の訓練データで微調整したモデルの性能を評価する。その 2 種類とは、人間が書いたデータと、QwQ-32B-Preview が生成した推論データである。 具体的には、Qwen2.5-32B を各データセット（データセットあたり 50 例）で微調整する。データセットには、人間が書いたデータの異なる段階（human_stage1 から human_stage4）と、QwQ-32B-Preview による合成データが含まれる。 得られたモデルは、いくつかの一般的な数学ベンチマークで評価され、結果は下の表に示す。 | データ | AIME24 | AIME25 | HMMT-24-25 | |-------------------------|-----------------|---------------|--------------| | human_stage1 | 9.27 (33.33) | 6.25 (20.00) | 4.82 (11.73) | | human_stage2 | 6.93 (20.00) | 5.83 (20.00) | 4.62 (11.22) | | human_stage3 | 6.41 (20.00) | 4.69 (20.00) | 3.37 (8.16) | | human_stage4 | 6.41 (26.67) | 5.26 (23.33) | 4.05 (9.69) | | **qwq_32b_preview** | 28.02 (56.67) | 21.61 (40.00) | 16.80 (30.10)|指標は pass@1（maj@64）として示す。ここで pass@1 は 64 回の生成にわたる平均精度であり、maj@64 は多数決の結果である。 評価の設定の詳細は論文を参照。
データセットの所有者:
NVIDIA Corporation
データセットの作成日:
06/01/2025
ライセンス・利用規約:
データのフィールド:
Nemotron-Math-HumanReasoning データセットには次のフィールドが含まれる:
problem: 数学問題の文。
solution: この問題の解答。人間が書いたもの（human）か、QwQ-32B-Preview で生成されたもの（automated）のいずれか。
solution_type: 「human_stageX」（X は 1 から 4 までの数字）または「qwq_32b_preview」。詳細は論文を参照。
expected_answer: 問題の正解（ground truth）。
データセットのサイズ
Nemotron-Math-HumanReasoning は、OpenMathReasoning データセットの 50 問の数学問題、人間が書いた 200 件の解答、そして QwQ-32B-Preview が生成した 50 件の追加の解答からなる。
データセットのサイズは 2.4 Mb である。
データ形式
テキストのみ。
参考文献:
論文 へのリンク
想定される用途:
言語モデルの訓練。
倫理的配慮:
NVIDIA は、信頼できる AI は共有の責任であると考えており、幅広い AI 応用の開発を可能にするためのポリシーと実践を確立している。このデータセットを利用規約に従って利用する際には、開発者は社内のモデルチームと協力して、このモデルが関連する業界や用途の要件を満たしているかを確認し、予期しない製品の誤用に対処すべきである。
セキュリティ上の脆弱性または NVIDIA の AI に関する懸念は、こちら から報告できる。
原文（英語）
Dataset accompanying the paper: The Challenge of Teaching Reasoning to LLMs Without RL or Distillation.
All experiments were conducted using NeMo-Skills.
Dataset Description:
Nemotron-Math-HumanReasoning is a compact dataset featuring human-written solutions to math problems, designed to emulate the extended reasoning style of models like DeepSeek-R1. These solutions are authored by students with deep experience in Olympiad-level mathematics. We provide multiple versions of the solutions and report on how training with this data compares to training on model-generated solutions from QwQ-32B-Preview. We hope this release supports further research into the key elements that contribute to effective "reasoning-style" solutions. The dataset is available for non-commercial use.
We evaluate the performance of models fine-tuned on two types of training data: human-written data and reasoning data generated by QwQ-32B-Preview. Specifically, we fine-tune Qwen2.5-32B on each dataset (50 examples per dataset), including different stages of human-curated data (human_stage1 to human_stage4) and synthetic data from QwQ-32B-Preview. The resulting models are evaluated on several popular mathematical benchmarks, and the results are reported in the table below.
| Data | AIME24 | AIME25 | HMMT-24-25 |
|---|---|---|---|
| human_stage1 | 9.27 (33.33) | 6.25 (20.00) | 4.82 (11.73) |
| human_stage2 | 6.93 (20.00) | 5.83 (20.00) | 4.62 (11.22) |
| human_stage3 | 6.41 (20.00) | 4.69 (20.00) | 3.37 (8.16) |
| human_stage4 | 6.41 (26.67) | 5.26 (23.33) | 4.05 (9.69) |
| qwq_32b_preview | 28.02 (56.67) | 21.61 (40.00) | 16.80 (30.10) |
We present metrics as pass@1 (maj@64) where pass@1 is an average accuracy across 64 generations and maj@64 is the result of majority voting. Please see our paper for more details on the evaluation setup.
Dataset Owner(s):
NVIDIA Corporation
Dataset Creation Date:
06/01/2025
License/Terms of Use:
Dataset fields:
Nemotron-Math-HumanReasoning dataset contains the following fields:
problem: Math problem statement.
solution: Solution to this problem. Either written by people (human) or generated with QwQ-32B-Preview (automated).
solution_type: “human_stageX” (X is a number from 1 to 4) or “qwq_32b_preview”. See our paper to learn more details.
expected_answer: The ground-truth answer to the problem.
Dataset Size
Nemotron-Math-HumanReasoning consists of 50 math problems from OpenMathReasoning dataset and 200 solutions written by humans, as well as 50 extra solutions generated by QwQ-32B-Preview.
The dataset size is 2.4 Mb.
Dataset Format
Text-only.
Reference(s):
Link to paper
Intended Usage:
Training language models.
Ethical Considerations:
NVIDIA believes Trustworthy AI is a shared responsibility and we have established policies and practices to enable development for a wide array of AI applications. When downloaded or used in accordance with our terms of service, developers should work with their internal model team to ensure this model meets requirements for the relevant industry and use case and addresses unforeseen product misuse.
Please report security vulnerabilities or NVIDIA AI Concerns here.
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