We denote by the set of all positive real numbers.
Find all functions which have the property: for all positive real numbers and .
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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 $y$.
Source: Međunarodna matematička olimpijada, shortlist 2005
Školjka 
the set of all positive real numbers.
which have the property: 
for all positive real numbers 
and 
.