Question 9
(POL 2) Let $a,b,c$ be positive numbers with $\sqrt a+\sqrt b+\sqrt c=\frac{\sqrt3}{2}$. Prove that the system of equations
$$\begin{aligned}\sqrt{y-a}+\sqrt{z-a}&=1,\\\sqrt{z-b}+\sqrt{x-b}&=1,\\\sqrt{x-c}+\sqrt{y-c}&=1,\end{aligned}$$
has exactly one solution $(x,y,z)$ in real numbers.
$$\begin{aligned}\sqrt{y-a}+\sqrt{z-a}&=1,\\\sqrt{z-b}+\sqrt{x-b}&=1,\\\sqrt{x-c}+\sqrt{y-c}&=1,\end{aligned}$$
has exactly one solution $(x,y,z)$ in real numbers.