Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
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](/media/m/c/3/e/c3e5b8e24544d937c6cc117debee1e02.png)
there is exactly one real number satisfying both of them. Determine all possible values of
![a^2+b^2+c^2](/media/m/d/8/6/d86474b88c5bdeccb25dff423e43ae61.png)
.
%V0
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 possible values of $a^2+b^2+c^2$.