DE-MO 2017 Problem 1

Given two real numbers $p$ and $q$ , we study the following system of equations with variables $x,y \in \mathbb{R}$ : \begin{align*} x^2+py+q&=0, y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$ .