ISL 1984

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations $$\sqrt{y-a}+\sqrt{z-a}=1$$ $$\sqrt{z-b}+\sqrt{x-b}=1$$ $$\sqrt{x-c}+\sqrt{y-c}=1$$ has exactly one solution $(x,y,z)$ in real numbers.

It was proposed by Poland. Have fun! :lol: