IMO 1959 - Problem 2

Problem 2: For which real number $x$ do the following equations hold:
(a) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2},$
(b) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1,$
(c) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=2,$
Solution: For the square roots to be real we must have $2x-1\ge 0\implies x\ge \frac{1}{2}$ and $x\ge \sqrt{2x-1}\implies (x-1)^2\ge 0,$ which always holds. Then we have $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=c\iff c^2=2x+2\sqrt{(x-1)^2}=2x+2|x-1|=\left\{\begin{array}{I} 2,\frac{1}{2}\le x\le 1,\\ 4x-2,x\ge 1\end{array}\right.$
(a) $c^2=2$. The equation holds for $\frac{1}{2}\le x\le 1$.
(b) $c^2=1$. The equation has no solution.
(c) $c^2=4$. The equation holds for $4x-2=4\implies x=\frac{3}{2}$.