APMO 2025 Problem 5

Consider an infinite sequence $a_{1},a_{2},\ldots$ of positive integers such that

$$100!(a_{m} + a_{m + 1} + \cdot \cdot \cdot +a_{n}) \mathrm{is~a~multiple~of} a_{n - m + 1}a_{n + m}$$

for all positive integers $m,n$ such that $m\leq n$

Prove that the sequence is either bounded or linear.

Observation: A sequence of positive integers is bounded if there exists a constant $N$ such that $a_{n}< N$ for all $n\in \mathbb{Z}_{>0}$ . A sequence is linear if $a_{n} = n\cdot a_{1}$ for all $n\in \mathbb{Z}_{>0}$