Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$ Find all the function that satisfy :
$$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$
I only find $f(0)=0$ but I can't prove $f(x)=2x$ 

If your $f$ is differentiable at $x=0$. Then
Differentiate the equation for $x$ at $x=0$
$f'\left(\dfrac{xf(y)}{2}\right)\dfrac{f(y)}{2}+f'\left(\dfrac{f(x)y}{2}\right)\dfrac{y}{2}f'(x)=4y$.
Take $x=0$. Since $f(0)=0$. Say $A = f'(0)$.
$A\dfrac{f(y)}{2}+A\dfrac{y}{2}A=4y$
Which is 
$f(y)= \dfrac{8y-A^2y}{A}$
$f'(y) = \dfrac{8-A^2}{A}$,take $y=0$, $f'(0)=A=\dfrac{8-A^2}{A}$.
$A=2$ or $A=-2$.
This is for smooth case. If not differentiable at $x=0$, then I have no idea.
