MEMO 2009 Problem 6

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