3. Find the relationship between the coefficients $a, b, c$ such that 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 a solution.
Solving: Add the equations in the system, we get
$$
\begin{array}{l}
(a+b+c) x^{2}+(a+b+c) x+(a+b \\
+c)=0 .
\end{array}
$$

Thus, $(a+b+c)\left(x^{2}+x+1\right)=0$.
If $a+b+c \neq 0$, then for any $x, x^{2} + x + 1 \neq 0$. Therefore, the equation has no solution. Hence, for the system of equations to have a solution, it must be that $a+b+c=0$. We prove that this condition is also sufficient. Indeed, if $a+b+c=0$, then $x=1$ is clearly a solution to the system of equations.