Question 150
Determine all triples $(x, y, z)$ of positive numbers satisfying the system of equations
$$
\begin{align*}
2x^3 &= 2y(x^2 + 1) - (z^2 + 1), \\
2y^4 &= 3z(y^2 + 1) - 2(x^2 + 1), \\
2z^5 &= 4x(z^2 + 1) - 3(y^2 + 1).
\end{align*}
$$
$$
\begin{align*}
2x^3 &= 2y(x^2 + 1) - (z^2 + 1), \\
2y^4 &= 3z(y^2 + 1) - 2(x^2 + 1), \\
2z^5 &= 4x(z^2 + 1) - 3(y^2 + 1).
\end{align*}
$$