AUSTRIAN-POLISH 1994 Problem 6

Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy: $$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 ... (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$ Show that there exists $j, 1 \le j \le n$ , such that $x_j = x_{j+1}$ . Here $x_{n+1} = x_1$ .