Question 4
Consider an infinite sequence $a_{1}, a_{2}, \ldots$ of positive integers such that
$$
100!\left(a_{m}+a_{m+1}+\cdots+a_{n}\right) \quad \text{ is a multiple of } a_{n-m+1} a_{n+m}
$$
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 $a_{n}\lt N$ for all $n \in \mathbb{Z}_{\gt 0}$. A sequence is linear if $a_{n}=n \cdot a_{1}$ for all $n \in \mathbb{Z}_{\gt 0}$.
$$
100!\left(a_{m}+a_{m+1}+\cdots+a_{n}\right) \quad \text{ is a multiple of } a_{n-m+1} a_{n+m}
$$
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 $a_{n}\lt N$ for all $n \in \mathbb{Z}_{\gt 0}$. A sequence is linear if $a_{n}=n \cdot a_{1}$ for all $n \in \mathbb{Z}_{\gt 0}$.