IMO Shortlist 1984 problem 9
Dodao/la:
arhiva2. travnja 2012. Let
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
be positive numbers with
![\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}](/media/m/7/0/5/7052385d260276c6d42a6b346a6504c3.png)
. Prove that the system of equations
![\sqrt{x-c}+\sqrt{y-c}=1](/media/m/1/9/9/1992ae7aa57c84aa2a8a8691bd3f42ec.png)
has exactly one solution
![(x, y, z)](/media/m/f/2/d/f2d4c9b9b3e7f29445f7a1063c15263f.png)
in real numbers.
%V0
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1984