10. (15th All-Russian Mathematical Olympiad, 1989) Find the relationship between the coefficients $a, b, c$ for which the system of equations
$$
\left\{\begin{array}{l}
a x^{2}+b x+c=0 \\
b x^{2}+c x+a=0 \\
c x^{2}+a x+b=0
\end{array}\right.
$$

has real solutions.
10. Adding the equations in the system, we get
$$
(a+b+c) x^{2}+(a+b+c) x+(a+b+c)=0,
$$

thus $(a+b+c)\left(x^{2}+x+1\right)=0$.
Since for any real number $x, x^{2}+x+1 \neq 0$, the necessary condition for the system of equations to have a solution is $a+b+c=0$.
Conversely, if $a+b+c=0$, then $x=1$ is clearly a solution to the system of equations.
Therefore, $a+b+c=0$ is also a sufficient condition for the system of equations to have a solution.