This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND
OR
NO

Found problems: 2

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 $g(n)>n^{0.999n}$.
asymptotic behaviour number theory p-adic valuation

2024 VJIMC, 3
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 \[\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.\]
limit asymptotic behaviour sequence real analysis recurrence relation limits recurrence relations sequences