Middle European Mathematical Olympiad 2009 Problem T-2

Let $a, b, c$ be real numbers such that for every two of the equations $$x^2 + ax + b = 0, \quad x^2 + bx + c = 0, \quad x^2 + cx + a = 0$$ there is exactly one real number satisfying both of them. Determine all the possible values of $a^2 + b^2 + c^2$.