AUSTRIAN-POLISH 1984 Problem 5

Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$ , find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and $$\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 ... a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}$$