Problem

Source: SMO senior 2024 Q5

Tags: Sequence, bounded, algebra



Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.