Question 54
Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations
$$
\begin{aligned}
x^{2} y + y^{2} z & = 1040 \\
x^{2} z + z^{2} y & = 260 \\
(x - y)(y - z)(z - x) & = -540
\end{aligned}
$$
$$
\begin{aligned}
x^{2} y + y^{2} z & = 1040 \\
x^{2} z + z^{2} y & = 260 \\
(x - y)(y - z)(z - x) & = -540
\end{aligned}
$$