Task 5. (20 points)

Find $x_{0}-y_{0}$, if $x_{0}$ and $y_{0}$ are the solutions to the system of equations:

$$
\left\{\begin{array}{l}
x^{3}-2023 x=y^{3}-2023 y+2020 \\
x^{2}+x y+y^{2}=2022
\end{array}\right.
$$
# Solution.

Rewrite the system as

$$
\left\{\begin{array}{l}
x^{3}-y^{3}+2023 y-2023 x=2020 \\
x^{2}+x y+y^{2}=2022
\end{array}\right.
$$

Let $x_{0}$ and $y_{0}$ be the solution to the system of equations. Then

$\left\{\begin{array}{l}x_{0}{ }^{3}-y_{0}{ }^{3}+2023 y_{0}-2023 x_{0}=2020, \\ x_{0}{ }^{2}+x_{0} y_{0}+y_{0}{ }^{2}=2022 .\end{array}\right.$

Rewrite the first equation of the system as

$\left\{\begin{array}{l}\left(x_{0}-y_{0}\right)\left(x_{0}^{2}+x_{0} y_{0}+y_{0}{ }^{2}\right)-2023\left(x_{0}-y_{0}\right)=2020, \\ x_{0}{ }^{2}+x_{0} y_{0}+y_{0}{ }^{2}=2022 ;\end{array}\right.$

$\left\{\begin{array}{l}2022\left(x_{0}-y_{0}\right)-2023\left(x_{0}-y_{0}\right)=2020, \\ x_{0}^{2}+x_{0} y_{0}+y_{0}^{2}=2022 ;\end{array}\right.$

Then $x_{0}-y_{0}=-2020$.

Answer. -2020.