Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has zero, one or more than one real solutions if $(b-1)^2-4ac$ is negative, equal to zero or positive respectively.
Problem
Source: IMO 1968 A3
Tags: algebra, system of equations, polynomial, IMO, IMO 1968, Discriminant