Question 5
Consider an infinite sequence $a_1,a_2,\ldots$ of positive integers such that
$$100!\,(a_m+a_{m+1}+\cdots+a_n)$$
is a multiple of $a_{n-m+1}a_{n+m}$ for all positive integers $m\le n$. Prove that the sequence is either bounded or linear.
A sequence is bounded if there is a constant $N$ with $a_n\lt N$ for every positive integer $n$. It is linear if $a_n=n\,a_1$ for every positive integer $n$.
$$100!\,(a_m+a_{m+1}+\cdots+a_n)$$
is a multiple of $a_{n-m+1}a_{n+m}$ for all positive integers $m\le n$. Prove that the sequence is either bounded or linear.
A sequence is bounded if there is a constant $N$ with $a_n\lt N$ for every positive integer $n$. It is linear if $a_n=n\,a_1$ for every positive integer $n$.