Baltic Way 1994 · Problem 3

Solution: The expression is well-defined only for $|x|, |y| \leq 1$ and we can assume that $x, y \geq 0$. Let $x = \cos \alpha$ and $y = \cos \beta$ for some $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. This reduces the expression to

$$\cos \alpha \cos \beta + \cos \alpha \sin \beta + \cos \beta \sin \alpha - \sin \alpha \sin \beta = \cos (\alpha + \beta) + \sin (\alpha + \beta) = \sqrt{2} \cdot \sin \left(\alpha + \beta + \frac{\pi}{4}\right)$$

which does not exceed $\sqrt{2}$. The equality holds when $\alpha + \beta + \frac{\pi}{4} = \frac{\pi}{2}$, for example when $\alpha = \frac{\pi}{4}$ and $\beta = 0$, i.e., $x = \frac{\sqrt{2}}{2}$ and $y = 1$.