Question 183
Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system
$$
\begin{aligned}
(a + b)(a^2 + b^2) &= (c + d)(c^2 + d^2), \\
(a + c)(a^2 + c^2) &= (b + d)(b^2 + d^2), \\
(a + d)(a^2 + d^2) &= (b + c)(b^2 + c^2).
\end{aligned}
$$
$$
\begin{aligned}
(a + b)(a^2 + b^2) &= (c + d)(c^2 + d^2), \\
(a + c)(a^2 + c^2) &= (b + d)(b^2 + d^2), \\
(a + d)(a^2 + d^2) &= (b + c)(b^2 + c^2).
\end{aligned}
$$