UA-UMO 1998 Problem 118

The sequence $(x_n)$ is given by $x_1=1$ and $$x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1.$$ (a) Prove that $x_{n+1}\ge x_n$ for all $n\ge4$ . (b) Prove that $\lfloor x_n\rfloor=n$ for all $n\ge4$ .