conversations listlengths 2 2 | dataset_index stringlengths 1 3 | completion_tokens int64 889 32.7k |
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"value": "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \\ldots, A_{k}$ of $X$ is tight if the union $A_{1} \\cup \\cdots \\cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. F... | 4 | 30,450 |
[
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"value": "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \\geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \\ldots, y_{k}... | 7 | 21,592 |
[{"from":"user","value":"Let $n$ be a positive integer and fix $2 n$ distinct points on a circumfere(...TRUNCATED) | 8 | 29,231 |
[{"from":"user","value":"Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such t(...TRUNCATED) | 11 | 16,794 |
[{"from":"user","value":"Find all positive integers $n$, for which the numbers in the set $S=\\{1,2,(...TRUNCATED) | 13 | 16,813 |
[{"from":"user","value":"Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divide(...TRUNCATED) | 15 | 23,749 |
[{"from":"user","value":"Find all surjective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ suc(...TRUNCATED) | 16 | 24,050 |
[{"from":"user","value":"Denote by $\\mathbb{Q}^{+}$the set of all positive rational numbers. Determ(...TRUNCATED) | 23 | 29,057 |
[{"from":"user","value":"Find the least positive integer $n$ for which there exists a set $\\left\\{(...TRUNCATED) | 26 | 31,296 |
[{"from":"user","value":"Determine the smallest number $M$ such that the inequality\n\n$$\n\\left|a (...TRUNCATED) | 29 | 27,061 |
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