BG-MO 1996 Problem 1

Sequence $\{a_n\}$ it define $a_1=1$ and $$a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$$ for all $n\ge 1$ Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$