TH-TMO 2015 Problem 1

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$ . Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$ .