PL-OM 1999 Problem 4

For which $n$ do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \cdots \cdots \cdots & \cdots & \cdots \cdots x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}