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.