Olympiad problems

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Found problems: 53

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2022 SEEMOUS, 4

Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$ is convergent.

2019 IMC, 7

Tags: number theory , convergence

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

1964 Putnam, B5

Tags: least common multiple , convergence

Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers. Prove that the series $$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$ is convergent

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