50th Mathematical Olympiad in Ukraine, Fourth Round (March 24, 2010)

We have $x\in [0,2]$, for such $x$ both functions $\sin \frac{\pi \sqrt{x}}{4}$ and $\cos \frac{\pi \sqrt{2-x}}{4}$ are increasing, thus, their sum is also increasing function. Hence, our equation has at most one real root. From the other hand, one can easily check that $x=1$ is, indeed, the solution.