Ukrajina, 2008

Fig.8

The graph of functions $y = |x - a| + |x - b|$ for $b > a$ is shown in fig.8. Thus it is clear that equation $|x - a| + |x - b| = c$ may have a single solution only if $a = b$ and $c = 0$ (fig.9).

Therefore for our equation the following condition should be satisfied:
$$\begin{cases} \sin \alpha = \frac{1}{2} \\ \cos 3\alpha = 0 \end{cases} \Rightarrow \begin{cases} \alpha = (-1)^n \frac{\pi}{6} + \pi n, & n, k \in \mathbb{Z} \\ \alpha = \frac{\pi}{6} + \frac{1}{3} \pi k \end{cases}$$
This system has the following common solution: $\alpha = (-1)^n \frac{\pi}{6} + \pi n, n \in \mathbb{Z}$ and $x = \frac{1}{2}$.

Fig.9