CWMO 2007 Problem 6

Find all natural numbers $n$ such that there exist $x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: $$x_1 + x_2 + \ldots + x_n = 0$$ $$ny^2 = x_1^2 + x_2^2 + \ldots + x_n^2$$