question stringlengths 40 2.42k | solution stringlengths 1 6.9k | cot_type stringclasses 3
values | source_type stringclasses 34
values | answer_magistral_small stringlengths 6.89k 159k |
|---|---|---|---|---|
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$ | 1. **Rewrite the function using trigonometric identities:**
\[
f(x) = 1 - a \cos(x) - b \sin(x) - A \cos(2x) - B \sin(2x)
\]
We can use the angle addition formulas for sine and cosine:
\[
\cos(x + \theta) = \cos(x)\cos(\theta) - \sin(x)\sin(\theta)
\]
\[
\sin(x + \theta) = \sin(x)\cos(\theta)... | math | AI-MO/NuminaMath-CoT/aops_forum | <think>
Alright, I have this problem:
**Problem Statement:**
Let \( a, b, A, B \) be given reals. We consider the function defined by
\[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \]
Prove that if for any real number \( x \) we have \( f(x) \geq 0 \) then \( a^2 + b^2 \leq 2 \)... |
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product? | 128 | math | qq8933/AIME_1983_2024 | <think>
Alright, I have a problem to solve:
**Problem Statement:**
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will \(20!\) be the resulting product?
### Understanding the Problem
Fi... |
Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$ . Show that there is a point $ y\in\mathcal{H}$ such that
\[ \lef... | 1. **Consider a countable subset \( S_0 \subseteq S \)**:
Let \( S_0 = \{x_1, x_2, \ldots\} \) be a countable subset of \( S \). Since \( S \) is a set of points in \( \mathcal{H} \) such that the distance between any two distinct points is \( d \), the same property holds for \( S_0 \).
2. **Find a point \( y \) f... | math | AI-MO/NuminaMath-CoT/aops_forum | <think>
Alright, I have this problem about an infinite-dimensional Hilbert space and a set of points with equal distances between any two distinct points. I need to show that there's a point y such that when I take each point x in S, subtract y from it, scale it by √2/d, and collect all these vectors, they form an orth... |
"Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_(...TRUNCATED) | 109 | math | qq8933/AIME_1983_2024 | "<think>\nAlright, I have this problem:\n\n**Find the remainder when \\(9 \\times 99 \\times 999 \\t(...TRUNCATED) |
"We randomly choose a function $f:[n] \\rightarrow[n]$, out of the $n^{n}$ possible functions. We al(...TRUNCATED) | "Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (in(...TRUNCATED) | math | KbsdJames/Omni-MATH | "<think>\nAlright, I have this probability problem to solve. Let me break it down and understand it (...TRUNCATED) |
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s | 167.0 | math | TIGER-Lab/TheoremQA/float | "<think>\nAlright, I have this problem: Compute the mean molecular speed \\( v \\) in the heavy gas (...TRUNCATED) |
"Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\(...TRUNCATED) | 1.3 | math | daman1209arora/jeebench/phy | "<think>\nAlright, I have a problem to solve:\n\n**Problem Statement:**\nTwo capacitors with capacit(...TRUNCATED) |
"One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpo(...TRUNCATED) | 181 | math | qq8933/AIME_1983_2024 | "<think>\nAlright, I have this problem about a trapezoid. Let me break it down and understand it ste(...TRUNCATED) |
"Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a p(...TRUNCATED) | "1. **Defining the Sparse Set \\( S \\) and the Turing Machine \\( M \\)**\n\n We need to construc(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "<think>\nAlright, I have this problem about sparse sets and oracle machines, and some variations of(...TRUNCATED) |
"In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The te(...TRUNCATED) | "(a) Let the number of promoted participants be \\( P \\) and the number of repeaters be \\( R \\). (...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "<think>\nAlright, I have this problem about a math test with N participants where N is less than 40(...TRUNCATED) |
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s1k-magistral-small-2506 Dataset
Overview
This dataset contains 1,000 mathematical reasoning problems from the simplescaling/s1K dataset, processed with Mistral's magistral-small-2506 model using their official reasoning system prompt. Each problem includes the original question, solution, metadata, and the model's reasoning response.
Dataset Structure
The dataset contains the following columns:
- question: The original mathematical problem or question
- solution: The ground truth solution to the problem
- cot_type: Chain-of-thought reasoning type classification
- source_type: The source or origin of the problem
- answer_magistral_small: The model's response using Mistral's reasoning framework
Source Data
This dataset is derived from simplescaling/s1K, which contains 1,000 carefully curated mathematical reasoning problems with solutions and metadata.
Model Configuration
- Model:
magistral-small-2506 - System Prompt: Official Mistral reasoning template requiring
<think>tags for internal reasoning followed by a concise summary - Processing: Parallel processing with exponential backoff for API reliability
- Error Handling: Automatic retries with graceful error recovery
Reasoning Format
Each model response follows Mistral's official reasoning template:
<think> [Internal reasoning and draft work, similar to scratch paper calculations] </think>
[Concise summary with clear final answer, formatted in Markdown]
Data Processing
- Source Dataset: simplescaling/s1K
- Processing Method: Concurrent API calls with 10 workers
- Checkpointing: Automatic checkpoint saving every 5 minutes for resume capability
- Quality Control: Exponential backoff retry mechanism for API reliability
Usage
This dataset is suitable for:
- Evaluating mathematical reasoning capabilities
- Studying chain-of-thought reasoning patterns
- Comparing model reasoning approaches
- Training reasoning-aware models
Technical Details
- Format: Parquet/Arrow format via Hugging Face Datasets
- Size: 1,000 examples
- Processing Time: Variable based on API response times
- Reliability: Built-in error handling and checkpoint recovery
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