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

$$
y f(2 x)-x f(2 y)=8 x y\left(x^{2}-y^{2}\right)
$$

## 1st solution
Set $y=1 / 2$ and $x=z / 2$, then follows

$$
\frac{f(z)}{2}-\frac{f(1) z}{2}=\frac{z^{3}-z}{2}
$$

With $c=f(1)-1$, $f(z)=z^{3}+c z$ therefore applies for all $z \in \mathbb{R}$. Substituting shows that these functions are solutions of the equation for all $c \in \mathbb{R}$.

## 2nd solution

With $y=0$, $f(0)=0$ follows. In addition, the equation for $x, y \neq 0$ can be transformed to

$$
\frac{f(2 x)}{x}-8 x^{2}=\frac{f(2 y)}{y}-8 y^{2} \quad \forall x, y \neq 0
$$

It follows immediately that $f(2 x) / x-8 x^{2}$ is constant for $x \neq 0$, for example equal to $c / 4$. Solving gives $f(x)=x^{3}+c x$ and because $f(0)=0$ this applies everywhere.