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.