ÖMO 2017 Problem 3

Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$ . Show that the sequence does not contain a square of a rational number.

Proposed by Theresia Eisenkölbl