22. A function $f$ satisfies the equation $(n-2019) f(n)-f(2019-n)=2019$ for every integer $n$.
What is the value of $f(2019)$ ?
A 0
B 1
C $2018 \times 2019$
D $2019^{2}$
E $2019 \times 2020$
Solution
C
Putting $n=0$ in the equation
$$
(n-2019) f(n)-f(2019-n)=2019
$$
gives
$$
-2019 f(0)-f(2019)=2019
$$
from which it follows that
$$
f(2019)=-2019 f(0)-2019 .
$$

Putting $n=2019$ in equation (1) gives
$$
-f(0)=2019
$$
and hence
$$
f(0)=-2019 \text {. }
$$

Substituting from (5) in (3) gives
$$
\begin{aligned}
f(2019) & =-2019 \times-2019-2019 \\
& =2019 \times 2019-2019 \\
& =2019(2019-1) \\
& =2019 \times 2018 .
\end{aligned}
$$

Therefore, the value of $f(2019)$ is $2018 \times 2019$.