5. Solution. The original equation is equivalent to

$$
(\sin 7 x+\sin 3 x)^{2}+(\cos 3 x+\cos x)^{2}+\sin ^{2} \pi a=0
$$
Answer: For $a \in \mathbb{Z}, x=\pi / 4+\pi k / 2, k \in \mathbb{Z}$, there are no solutions for other $a$.

## Lomonosov Moscow State University

 Olympiad "Conquer Sparrow Hills" Variant 4-1 (Cheboksary)