Thai Mathematical Olympiad

There are many functions $f$ that satisfy the given condition. One of them is given by the following definition: $$f(n)=\{\begin{array}{ll}n& \text{if }n\ge 0\\ -3n& \text{if }n<0\end{array}$$ which can be verified in the following 3 cases.

$n>0$: $f(f(n)-2n)=f(-n)=3n$ and $2f(n)+n=2n+n=3n$.

$n<0$: $f(f(n)-2n)=f(-5n)=-5n$ and $2f(n)+n=-5n$.

$n=0$: We have $f(0)=0$ and $f(f(n)-2n)=2f(n)+n=0$.