## Task B-4.1.

Solve the system of equations

$$
\left\{\begin{aligned}
\sin \frac{\pi x}{2022}-\sin \frac{\pi y}{2022} & =1 \\
x-y & =2022
\end{aligned}\right.
$$

if $|x| \leqslant 2022$ and $|y| \leqslant 2022$.
## Solution.

If we express $x=2022+y$ from the second equation and substitute it into the first equation, we get

$$
\sin \left(\pi+\frac{\pi y}{2022}\right)-\sin \frac{\pi y}{2022}=1
$$

Since $\sin \left(\pi+\frac{\pi y}{2022}\right)=-\sin \frac{\pi y}{2022}$,

the given equation simplifies to the basic equation $\sin \frac{\pi y}{2022}=-\frac{1}{2}$.

From this, we have $\frac{\pi y_{1}}{2022}=-\frac{\pi}{6}+2 k \pi$ and $\frac{\pi y_{2}}{2022}=\frac{7 \pi}{6}+2 k \pi, k \in \mathbb{Z}$.

After rearranging, we get

$y_{1}=-337+4044 k$ and $y_{2}=2359+4044 k, k \in \mathbb{Z}$, and then

$x_{1}=2022+y_{1}=1685+4044 k$ and $x_{2}=2022+y_{2}=4381+4044 k, k \in \mathbb{Z}$.

Therefore, the solutions to the given system are

$(1685+4044 k,-337+4044 k)$ and $(4381+4044 k, 2359+4044 k), k \in \mathbb{Z}$, but we need to apply the conditions

$|x| \leqslant 2022,|y| \leqslant 2022$, or $-2022 \leqslant x \leqslant 2022,-2022 \leqslant y \leqslant 2022$.

By substituting $k=0$ into the first solution $(1685+4044 k,-337+4044 k)$, we get the solution $(1685,-337)$.

By substituting $k=-1$ into the second solution $(4381+4044 k, 2359+4044 k)$, we get the solution $(337,-1685)$.

Finally, the solutions are $(1685,-337)$ and $(337,-1685)$.