Olympiad problems

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2023 IMC, 10

For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that \[1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.\] Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$.

2024 VJIMC, 3

Tags: recurrence relation , asymptotic behaviour , sequence , real analysis , limit

Let $a_1>0$ and for $n \ge 1$ define \[a_{n+1}=a_n+\frac{1}{a_1+a_2+\dots+a_n}.\] Prove that \[\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.\]

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