KZ-MO 2003 Problem 1

Find all natural numbers $n$ ,such that there exist $x_1,x_2,\dots,x_{n+1}\in\mathbb{N}$ ,such that $\frac{1}{x_1^2}+\frac{1}{x_2^2}+\dots+\frac{1}{x_n^2}=\frac{n+1}{x_{n+1}^2}$ .