Question 4
(BUL 2), IMO 3. Let $a,b,c$ be real numbers. Prove that the system of equations
$$\begin{cases}ax_1^2+bx_1+c=x_2,\\ax_2^2+bx_2+c=x_3,\\\cdots\\ax_{n-1}^2+bx_{n-1}+c=x_n,\\ax_n^2+bx_n+c=x_1.\end{cases}$$
has a unique real solution if and only if $(b-1)^2-4ac=0$.
Remark. It is assumed that $a\ne0$.
$$\begin{cases}ax_1^2+bx_1+c=x_2,\\ax_2^2+bx_2+c=x_3,\\\cdots\\ax_{n-1}^2+bx_{n-1}+c=x_n,\\ax_n^2+bx_n+c=x_1.\end{cases}$$
has a unique real solution if and only if $(b-1)^2-4ac=0$.
Remark. It is assumed that $a\ne0$.