id large_stringlengths 24 24 | competition large_stringclasses 22
values | year large_stringdate 1959-01-01 00:00:00 2024-01-01 00:00:00 | problem large_stringlengths 36 1.6k | trace large_stringlengths 2.75k 118k | solution large_stringlengths 1.09k 44.3k |
|---|---|---|---|---|---|
6733856f52d7e82fe11d4828 | Kosovo National Mathematical Olympiad | 2011 | Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
$$ a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}
$$
In what case does equality hold? | Okay, let's try to prove that for any triangle with sides \(a\), \(b\), \(c\) and area \(S\), the inequality \(a^2 + b^2 + c^2 \geq 4S\sqrt{3}\) holds, and find when equality occurs. Hmm, first, I remember that for triangles, there are several formulas related to the area, like Heron's formula, or using \(S = \frac{1}{... | To prove the inequality
$$
a^2 + b^2 + c^2 \geq 4S\sqrt{3}
$$
for any triangle with sides $a$, $b$, $c$ and area $S$, we proceed through a combination of trigonometric identities, geometric reasoning, and algebraic manipulation.
---
### **Step 1: Expressing the Area Using Trigonometry**
Let’s consider the angle ... |
6733857b52d7e82fe11d60b5 | Austrian-Polish | 1980 | Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1} \lt a_{2} \lt \ldots \lt a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: $$x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.$$ | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so the problem is to find the greatest natural number n where we have natural numbers x₁ to xₙ, and natural numbers a₁ < a₂ < ... < aₙ₋₁ such that two conditions hold: first, the produ... | To solve the problem, we are given a natural number $ N = 1980 $ and asked to find the **greatest natural number** $ n $ such that there exist natural numbers $ x_1, x_2, \ldots, x_n $ and strictly increasing natural numbers $ a_1 < a_2 < \cdots < a_{n-1} $ satisfying:
$$
x_1 x_2 \cdots x_n = 1980 \quad \text{and} \qu... |
6733857b52d7e82fe11d6115 | Azerbaijan Team Selection Test | 2020 | You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$. | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so the problem says we have n blocks, each at least weight 1, total weight 2n. We need to show that for any r between 0 and 2n-2 inclusive, there's a subset with weight between r and r... | To solve the problem, we are given $ n $ blocks, each with weight at least 1, and their total weight is $ 2n $. We are to prove that for every real number $ r $ satisfying $ 0 \leq r \leq 2n - 2 $, there exists a subset of these blocks whose total weight lies in the interval $ [r, r + 2] $.
---
### **Key Observations... |
6733857b52d7e82fe11d6114 | Azerbaijan Team Selection Test | 2020 | Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying $$u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.$$Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
$$
a b \leqslant-\frac{1}{2019}.
... | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so we have 2019 real numbers that add up to zero, their squares add up to 1, and we need to show that the product of the minimum and maximum is at most -1/2019. Hmm, okay, let's start ... | We are given a sequence of real numbers $ u_1, u_2, \dots, u_{2019} $ satisfying:
$$
u_1 + u_2 + \cdots + u_{2019} = 0 \quad \text{and} \quad u_1^2 + u_2^2 + \cdots + u_{2019}^2 = 1,
$$
with $ a = \min(u_1, u_2, \ldots, u_{2019}) $ and $ b = \max(u_1, u_2, \ldots, u_{2019}) $. We are to prove that:
$$
ab \leq -\frac... |
6733857b52d7e82fe11d61b1 | Belarus Team Selection Test | 2014 | "Let $\\mathbb{Z} _{ \\gt 0}$ be the set of positive integers. Find all functions $f: \\mathbb{Z} _(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "To solve the functional equation:\n\n$$\nm^2 + f(n) \\mid mf(m) + n \\quad \\text{for all } m, n \\(...TRUNCATED) |
6733858352d7e82fe11d6a23 | Germany Team Selection Test | 2022 | "Find all positive integers $n\\geq1$ such that there exists a pair $(a,b)$ of positive integers, su(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "To solve the problem, we are to find all positive integers $ n \\geq 1 $ such that there exists a p(...TRUNCATED) |
6733858752d7e82fe11d6ada | Germany Team Selection Test | 2007 | "Prove the inequality:\n$$\\sum_{i \\lt j}{\\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\\leq \\frac {n}{2(a(...TRUNCATED) | "Okay, let's try to tackle this inequality. The problem is to prove that for positive real numbers \(...TRUNCATED) | "To prove the inequality\n\n$$\n\\sum_{i < j} \\frac{a_i a_j}{a_i + a_j} \\leq \\frac{n}{2(a_1 + a_2(...TRUNCATED) |
6733858752d7e82fe11d6b18 | Germany Team Selection Test | 2004 | "Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and nega(...TRUNCATED) | "Okay, let's try to tackle this problem. So, we have a 3x3 matrix with positive diagonal entries and(...TRUNCATED) | "To prove the existence of positive real numbers $ c_1, c_2, c_3 $ such that the vector \n$$\n\\beg(...TRUNCATED) |
6733858752d7e82fe11d6b3d | Germany Team Selection Test | 1978 | "Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two su(...TRUNCATED) | "Okay, let's try to tackle this problem step by step. First, let's make sure I understand the proble(...TRUNCATED) | "To solve the problem, we consider a set $ B $ of $ k $ sequences, each of length $ n $ with entries(...TRUNCATED) |
6733858752d7e82fe11d6b3e | Germany Team Selection Test | 1977 | "We consider two sequences of real numbers $x_{1} \\geq x_{2} \\geq \\ldots \\geq x_{n}$ and $\\ y_{(...TRUNCATED) | "Okay, let's try to tackle this problem. So, we have two non-increasing sequences, x₁ ≥ x₂ ≥(...TRUNCATED) | "To prove the inequality \n$$\n\\sum_{i=1}^{n} (x_i - y_i)^2 \\leq \\sum_{i=1}^{n} (x_i - z_i)^2,\n(...TRUNCATED) |
End of preview. Expand in Data Studio
proof-correctness-bench
768 chain-of-thought reasoning traces for olympiad mathematics problems, generated by a 4B-parameter student model (Qwen3-4B family) and verified correct on the first attempt — no hints, no interventions, no trace editing. These are the fully self-generated subset of a larger scaffolded-generation corpus: an LLM verifier checked each final answer against a reference solution, and only first-pass successes are included here.
Fields
| Field | Description |
|---|---|
id |
stable trace id |
competition |
source competition (22 competitions) |
year |
competition year (1959–2024) |
problem |
problem statement |
trace |
the model's full chain-of-thought reasoning |
solution |
the model's final written solution |
Notes
- Traces average ~24k characters; solutions ~3k.
- Problem statements originate from public mathematics competitions.
- Traces and solutions are model-generated and released CC-BY-4.0.
- Verification was LLM-based (repeated-attempt answer checking against reference solutions); no human grading. Expect some residual noise.
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