Question 192
Find all the $(x, y, z)$ positive real triples that meet the $$
2 x \sqrt{x+1}-y(y+1)=1
$$ system $$
\begin{aligned}
& 2 y \sqrt{y+1}-z(z+1)=1 \\
& 2 z \sqrt{z+1}-x(x+1)=1
\end{aligned}
$$
2 x \sqrt{x+1}-y(y+1)=1
$$ system $$
\begin{aligned}
& 2 y \sqrt{y+1}-z(z+1)=1 \\
& 2 z \sqrt{z+1}-x(x+1)=1
\end{aligned}
$$