Find all functions that satisfy the conditions and .
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Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
$$f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},$$
and $f(-1) \neq 0$.
Izvor: Međunarodna matematička olimpijada, shortlist 2012
Školjka 
that satisfy the conditions
.