Problem

Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade

Tags: number theory, Sequence



Let $(a_n)_{n\geq 1}$ be a (not necessarily strictly) increasing sequence of positive integers, such that $a_n \leq 1000n^{0.999}$ for every positive integer $n$. Prove that there exist infinitely many positive integers $n$ for which $a_n$ divides $n$.