CZ-SK-MO 2020 Problem 3

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, y^2 - 3z + p = x, z^2 - 3x + p = y \end{cases}$ with real parameter $p$ . a) For $p \ge 4$ , solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$ .

(Jaroslav Svrcek)