Question 32
Let $a_1, a_2, \dots$ be a sequence of positive numbers satisfying, for any positive integers $k, l, m, n$ such that $k + n = m + l$,
$$
\frac{a_k + a_n}{1 + a_k a_n} = \frac{a_m + a_l}{1 + a_m a_l}.
$$
Show that there exist positive numbers $b, c$ so that $b \le a_n \le c$ for any positive integer $n$.
$$
\frac{a_k + a_n}{1 + a_k a_n} = \frac{a_m + a_l}{1 + a_m a_l}.
$$
Show that there exist positive numbers $b, c$ so that $b \le a_n \le c$ for any positive integer $n$.