5. Let $a=\sin x, b=\cos y$. We obtain $\left\{\begin{array}{l}a^{2} b^{2}-2 a+b^{2}=0, \\ 2 a^{2}-4 a+b^{3}+3=0 .\end{array}\right.$ Consider the equations in the system as quadratic in terms of $a$. In the first equation, the discriminant is $4-4 b^{2}$, in the second equation, it is $-8-8 b^{3}$. The discriminants must be non-negative, so the only $b$ that can be a solution to the system is $b=-1$. Then the system has a unique solution $a=1, b=-1$.
Answer: $(\pi / 2 + 2 \pi n ; \pi + 2 \pi k), n, k \in Z$.

Answer to the option: $4-2: (2 \pi n ; 3 \pi / 2 + 2 \pi k), n, k \in Z$.

## Lomonosov Moscow State University
## Olympiad "Conquer Sparrow Hills" Variant 5-1