Question 4
Find all quadruples $(x_1,x_2,x_3,x_4)$ of real numbers satisfying the following six equalities:
$$
\begin{aligned}
x_1+x_2&=x_3^2+x_4^2+6x_3x_4,\\
x_1+x_3&=x_2^2+x_4^2+6x_2x_4,\\
x_1+x_4&=x_2^2+x_3^2+6x_2x_3,\\
x_2+x_3&=x_1^2+x_4^2+6x_1x_4,\\
x_2+x_4&=x_1^2+x_3^2+6x_1x_3,\\
x_3+x_4&=x_1^2+x_2^2+6x_1x_2.
\end{aligned}
$$
$$
\begin{aligned}
x_1+x_2&=x_3^2+x_4^2+6x_3x_4,\\
x_1+x_3&=x_2^2+x_4^2+6x_2x_4,\\
x_1+x_4&=x_2^2+x_3^2+6x_2x_3,\\
x_2+x_3&=x_1^2+x_4^2+6x_1x_4,\\
x_2+x_4&=x_1^2+x_3^2+6x_1x_3,\\
x_3+x_4&=x_1^2+x_2^2+6x_1x_2.
\end{aligned}
$$