US-TST 2001 Problem 1

Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$ . Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$ .