Given $a,b,c$ positive numbers, solve the system $\sqrt{xy}+\sqrt{xz}-x=a$, $\sqrt{yz}+\sqrt{yx}-y=b$ and $\sqrt{zx}+\sqrt{zy}-z=c$,
where $x,y,z\in \mathbb{R}$.
This only a pretty question. I did it with many changes of variables and now I want to see another solution for it.
My solution begins isolating $\sqrt{x}$ in the first equation, isolating $\sqrt{y}$ and $\sqrt{z}$ in the second and third one, respectively. After, the terms in each of three equations are symmetric, let $S_x=\sqrt{z}+\sqrt{y}-\sqrt{x}$ and so on. There are many relations among $S_x,S_y, S_y$. It will be important make $S_a=b+c-a$ and so on and $S=a+b+c$. The solution will be given in term of $S_a, S_b, S_c$ and $S$.
I want to know if someone can give another solution. Thanks!