Folklore

Find all non-negative solutions of the system of equations:

$$
\begin{aligned}
& x^{3}=2 y^{2}-z \\
& y^{3}=2 z^{2}-x \\
& z^{3}=2 x^{2}-y
\end{aligned}
$$
By adding all the equations of the system, we get $x(x-1)^{2}+y(y-1)^{2}+z(z-1)^{2}=0$. From this, $x(x-1)^{2}=y(y-1)^{2}=z(z-1)^{2}=0$, which means that the value of each of the unknowns can be 1 or 0 (in particular, all solutions are integers). Now, from the system, it is clear that $x, y$ and $z$ are of the same parity. Therefore, there are no more than two solutions: $(0,0,0)$ and $(1,1,1)$. By substitution, we verify that both fit.

## Answer