Openthoughts 114K

Name: Openthoughts 114K
Creator: Helix
Keywords: dataset, format:parquet, hugging face, library:dask, library:datasets, license:apache-2.0, modality:text, size_categories:100K<n<1M

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[!NOTE] We have released a paper for OpenThoughts! See our paper here. Open-Thoughts-114k Open synthetic reasoning dataset with 114k high-quality examples covering math, science, code, and puzzles! Inspect the content with rich formatting with Curator Viewer. Available Subsets default subset containing ready-to-train data used to finetune the OpenThinker-7B and OpenThinker-32B models: ds = load_dataset("open-thoughts/OpenThoughts-114k"… See the full description on the dataset page: https://huggingface.co/datasets/open-thoughts/OpenThoughts-114k.

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count of records by domain distribution of domain most common values in problem length distribution of problem

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#	problem text	deepseek_reasoning text	deepseek_solution text	ground_truth_solution text	domain text	source text
1	Let $ a, b, c $ be positive real numbers. Prove that $$ \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}, $$ and…	Okay, so I need to prove this inequality: $\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}$, where $a, b, c$ ar…	To prove the inequality \[ \frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc} \] for positive real numbers \(a,…	1. Consider the given inequality: \[ \frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc} \] To simplify, we add \…	math	numina_math
2	Given real numbers $ a, b, c $ and a positive number $ \lambda $ such that the polynomial $ f(x) = x^3 + a x^2 + b x + c $ has three …	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ …	Given real numbers $a, b, c$ and a positive number $\lambda$ such that the polynomial $f(x) = x^3 + ax^2 + bx + c$ has three real roo…	We begin by analyzing the function $ f(x) = x^3 + a x^2 + b x + c $, which has three real roots $ x_1, x_2, x_3 $. We are given the fo…	math	numina_math
3	In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share…	Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem says: In triangle ABC, CA equals CB, so i…	In triangle $ABC$, $CA = CB$, making it isosceles with $D$ as the midpoint of $AB$. Line $EF$ passes through $D$ such that tria…	1. Identify Key Elements: Consider $ \triangle ABC $ where $ CA = CB $ and $ D $ is the midpoint of $ AB $. Line \( \mathrm{EF}…	math	numina_math
4	Let $ p = 2^{3009}, q = 3^{2006}, $ and $ r = 5^{1003} $. Which of the following statements is true? (A) $ p < q < r $ (B) \( p < r <…	Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and…	To determine which of the given statements is true, we need to compare the sizes of $ p = 2^{3009} $, $ q = 3^{2006} $, and \( r = 5^{1…	Given the values: \[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \] 1. Express $ p $ and $ q $ in terms of powers of the same…	math	numina_math
5	When $ a < -1 $, the nature of the roots for the equation $$ \left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0 $$ is: (A) Two neg…	Okay, so I need to figure out the nature of the roots for the quadratic equation $(a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0$ when \(a < -1\…	To determine the nature of the roots for the quadratic equation $(a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0$ when $a < -1$, we analyze the…	Given the quadratic equation: \[ (a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0 \] where $ a < -1 $. 1. **Identify the coefficients and an…	math	numina_math
6	Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$.…	Okay, let's see. So, we have four points A, B, C, D in a plane. Then there are midpoints of the segments AB, AC, AD, BC, BD, CD, named Q, R…	Given four points $A, B, C, D$ in the plane, we consider the midpoints $Q, R, S, T, U, V$ of the segments $AB, AC, AD, BC, BD, CD$ re…	To find the distance between the midpoints of $Q U$ and $R V$, let's break down the given information and analyze each step. 1. **Identify…	math	numina_math
7	Given that $1 \leq x, y, z \leq 6$, how many cases are there in which the product of natural numbers $x, y, z$ is divisible by 10?	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 an…	To determine the number of ordered triples $(x, y, z)$ where $1 \leq x, y, z \leq 6$ and their product is divisible by 10, we use the p…	Given the constraints $1 \leq x, y, z \leq 6$, we are to find the number of natural number combinations $(x, y, z)$ such that their pro…	math	numina_math
8	Find all prime numbers $ p $ such that for any prime number $ q < p $, if $ p = kq + r $ with $ 0 \leq r < q $, then there does not…	Okay, let's try to tackle this problem. So, we need to find all prime numbers $ p $ such that for any prime number $ q < p $, when we d…	To find all prime numbers $ p $ such that for any prime number $ q < p $, if $ p = kq + r $ with $ 0 \leq r < q $, then there does …	Step 1: Identify the prime numbers $ p $ that meet the given conditions: For every prime $ q < p $, $ p = k q + r $ where \( 0…	math	numina_math
9	Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ that satisfy for all $ x, y \in \mathbb{R} $: \[ f(y-f(x)) = f(x) - 2x + …	Okay, so I need to find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ that satisfy the functional equation: \[ f(y - f(x)) = f…	To find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ that satisfy the functional equation: \[ f(y - f(x)) = f(x) - 2x + f(f(y…	Nous voulons trouver toutes les fonctions $ f: \mathbb{R} \rightarrow \mathbb{R} $ qui satisfont l'équation fonctionnelle suivante pour …	math	numina_math
10	Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ …	Okay, so I need to prove that triangle ABC is isosceles given some conditions about the perimeters of smaller triangles. Let me try to unde…	Given triangle $ABC$ with points $M$ on $AB$ and $N$ on $BC$, we know the perimeters of triangles $AMC$ and $CNA$ are equal, …	1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities: \[ P(\tr…	math	numina_math
11	Let $ a, b, c $ be given complex numbers, with $ \|a+b\| = m $ and $ \|a-b\| = n $. Given that $ m \cdot n \neq 0 $, prove that: $$ \ma…	Okay, so I need to prove that the maximum of \|ac + b\| and \|a + bc\| is at least mn over the square root of (m² + n²), where m is \|a + b\| and…	Given complex numbers $a, b, c$ with $\|a + b\| = m$ and $\|a - b\| = n$ where $m \cdot n \neq 0$, we need to prove that: $$ \max \{\|ac…	Given that $ a, b, c $ are complex numbers with $ \|a+b\|=m $ and $ \|a-b\|=n $, and $ mn \neq 0 $, we need to prove: \[ \max \{\|ac+b\|,…	math	numina_math
12	Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly…	Okay, so I need to find the limit of the probability p_n as n approaches infinity. The problem is about writing the first n natural numbers…	To determine the limit of the probability $ p_n $ as $ n $ approaches infinity, we need to analyze the distribution of the digit 0 in t…	1. Define Variables and Notation: Let's analyze the problem by denoting the number of digits in the number $n$ by $k$. Each of …	math	numina_math

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deepseek_reasoning text
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ground_truth_solution text
domain categorical
source categorical
test_cases categorical
starter_code categorical

#	problem text	deepseek_reasoning text	deepseek_solution text	ground_truth_solution text	domain text	source text
1	Let \( a, b, c \) be positive real numbers. Prove that $$ \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}, $$ and…	Okay, so I need to prove this inequality: \(\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}\), where \(a, b, c\) ar…	To prove the inequality \[ \frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc} \] for positive real numbers \(a,…	1. Consider the given inequality: \[ \frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc} \] To simplify, we add \…	math	numina_math
2	Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three …	Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ …	Given real numbers \(a, b, c\) and a positive number \(\lambda\) such that the polynomial \(f(x) = x^3 + ax^2 + bx + c\) has three real roo…	We begin by analyzing the function \( f(x) = x^3 + a x^2 + b x + c \), which has three real roots \( x_1, x_2, x_3 \). We are given the fo…	math	numina_math
3	In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share…	Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem says: In triangle ABC, CA equals CB, so i…	In triangle \(ABC\), \(CA = CB\), making it isosceles with \(D\) as the midpoint of \(AB\). Line \(EF\) passes through \(D\) such that tria…	1. Identify Key Elements: Consider \( \triangle ABC \) where \( CA = CB \) and \( D \) is the midpoint of \( AB \). Line \( \mathrm{EF}…	math	numina_math
4	Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true? (A) \( p < q < r \) (B) \( p < r <…	Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and…	To determine which of the given statements is true, we need to compare the sizes of \( p = 2^{3009} \), \( q = 3^{2006} \), and \( r = 5^{1…	Given the values: \[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \] 1. Express \( p \) and \( q \) in terms of powers of the same…	math	numina_math
5	When \( a < -1 \), the nature of the roots for the equation $$ \left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0 $$ is: (A) Two neg…	Okay, so I need to figure out the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\…	To determine the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\), we analyze the…	Given the quadratic equation: \[ (a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0 \] where \( a < -1 \). 1. **Identify the coefficients and an…	math	numina_math
6	Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$.…	Okay, let's see. So, we have four points A, B, C, D in a plane. Then there are midpoints of the segments AB, AC, AD, BC, BD, CD, named Q, R…	Given four points \(A, B, C, D\) in the plane, we consider the midpoints \(Q, R, S, T, U, V\) of the segments \(AB, AC, AD, BC, BD, CD\) re…	To find the distance between the midpoints of $Q U$ and $R V$, let's break down the given information and analyze each step. 1. **Identify…	math	numina_math
7	Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10?	Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 an…	To determine the number of ordered triples \((x, y, z)\) where \(1 \leq x, y, z \leq 6\) and their product is divisible by 10, we use the p…	Given the constraints \(1 \leq x, y, z \leq 6\), we are to find the number of natural number combinations \((x, y, z)\) such that their pro…	math	numina_math
8	Find all prime numbers \( p \) such that for any prime number \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), then there does not…	Okay, let's try to tackle this problem. So, we need to find all prime numbers \( p \) such that for any prime number \( q < p \), when we d…	To find all prime numbers \( p \) such that for any prime number \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), then there does …	Step 1: Identify the prime numbers \( p \) that meet the given conditions: For every prime \( q < p \), \( p = k q + r \) where \( 0…	math	numina_math
9	Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \): \[ f(y-f(x)) = f(x) - 2x + …	Okay, so I need to find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation: \[ f(y - f(x)) = f…	To find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation: \[ f(y - f(x)) = f(x) - 2x + f(f(y…	Nous voulons trouver toutes les fonctions \( f: \mathbb{R} \rightarrow \mathbb{R} \) qui satisfont l'équation fonctionnelle suivante pour …	math	numina_math
10	Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ …	Okay, so I need to prove that triangle ABC is isosceles given some conditions about the perimeters of smaller triangles. Let me try to unde…	Given triangle \(ABC\) with points \(M\) on \(AB\) and \(N\) on \(BC\), we know the perimeters of triangles \(AMC\) and \(CNA\) are equal, …	1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities: \[ P(\tr…	math	numina_math
11	Let \( a, b, c \) be given complex numbers, with \( \|a+b\| = m \) and \( \|a-b\| = n \). Given that \( m \cdot n \neq 0 \), prove that: $$ \ma…	Okay, so I need to prove that the maximum of \|ac + b\| and \|a + bc\| is at least mn over the square root of (m² + n²), where m is \|a + b\| and…	Given complex numbers \(a, b, c\) with \(\|a + b\| = m\) and \(\|a - b\| = n\) where \(m \cdot n \neq 0\), we need to prove that: $$ \max \{\|ac…	Given that \( a, b, c \) are complex numbers with \( \|a+b\|=m \) and \( \|a-b\|=n \), and \( mn \neq 0 \), we need to prove: \[ \max \{\|ac+b\|,…	math	numina_math
12	Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly…	Okay, so I need to find the limit of the probability p_n as n approaches infinity. The problem is about writing the first n natural numbers…	To determine the limit of the probability \( p_n \) as \( n \) approaches infinity, we need to analyze the distribution of the digit 0 in t…	1. Define Variables and Notation: Let's analyze the problem by denoting the number of digits in the number \(n\) by \(k\). Each of …	math	numina_math

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