1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$

$$
f(x f(y)-y f(x))=f(x y)-x y
$$

(Dušan Djukić)
1. Substituting $y=0$ yields $f(x f(0))=f(0)$. If $f(0) \neq 0$, the expression $x f(0)$ takes all real values, so $f$ is a constant function, which cannot be a solution. Therefore $f(0)=0$.

Setting $y=x$ gives us $f(0)=f\left(x^{2}\right)-x^{2}$, i.e. $f\left(x^{2}\right)=x^{2}$. Thus $f(x)=x$ for all $x \geqslant 0$. Next, we set $x, y0>y$ have already been verified, while for $x>0>y$ we have $-2 x y=f(-2 x y)=$ $f(x f(y)-y f(x))=f(x y)-x y=-2 x y$.

Therefore the only solutions are $f(x)=x$ and $f(x)=|x|$.