We denote by
![\mathbb{R}^+](/media/m/4/d/d/4dd6182efc1bb170a565248a692ee278.png)
the set of all positive real numbers.
Find all functions
![f: \mathbb R^ + \rightarrow\mathbb R^ +](/media/m/b/3/d/b3d2d5805489ef9814d45a4c21b75684.png)
which have the property:
![f\left(x\right)f\left(y\right) = 2f\left(x + yf\left(x\right)\right)](/media/m/a/2/5/a259791c78dda440f444ee5cd9e5b435.png)
for all positive real numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
.
%V0
We denote by $\mathbb{R}^+$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property: $$f\left(x\right)f\left(y\right) = 2f\left(x + yf\left(x\right)\right)$$ for all positive real numbers $x$ and $y$.