Question 161
Find all positive integers $n$ such that there exist non-zero integers $x_1, x_2, \dots, x_n, y$, satisfying the following conditions
$$
\begin{cases} x_1 + \cdots + x_n = 0, \\ x_1^2 + \cdots + x_n^2 = n y^2. \end{cases}
$$
$$
\begin{cases} x_1 + \cdots + x_n = 0, \\ x_1^2 + \cdots + x_n^2 = n y^2. \end{cases}
$$