IMO 1988 - Shortlisted problems Question 28

(GBR 5) The sequence $\{a_n\}$ of integers is defined by $a_1=2$, $a_2=7$, and $$-\frac12\lt a_{n+1}-\frac{a_n^2}{a_{n-1}}\le\frac12,$$ for $n\ge2$. Prove that $a_n$ is odd for all $n\gt 1$.