International Mathematical Olympiad 1968 Problem 3

Consider the system of equations $$\begin{aligned} ax_1^2 + bx_1 + c &= x_2 \\ ax_2^2 + bx_2 + c &= x_3 \\ &\vdots \\ ax_{n-1}^2 + bx_{n-1} + c &= x_n \\ ax_n^2 + bx_n + c &= x_1, \end{aligned}$$ with unknowns $x_1, x_2, \ldots, x_n$, where $a, b, c$ are real and $a \neq 0$. Let $\Delta = (b-1)^2 - 4ac$. Prove that for this system

(a) if $\Delta < 0$, there is no solution,

(b) if $\Delta = 0$, there is exactly one solution,

(c) if $\Delta > 0$, there is more than one solution.