Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$, we have

$$
f(x) f(y)+f(x+y)=x y
$$

(PAMO 2013)
Setting $x=0$, we find $f(0)=-1$ or $f(x=0)$ for all $x \in \mathbb{R}^{*}$, the latter possibility being quickly ruled out.

With $x=-y=1$, we find $f(1)=0$ or $f(-1)=0$. In the first case, we take $x=a-1$ and $y=1$ to find $f(a)=a-1$. In the second, we have analogously $f(a)=a+1$. Conversely, these two solutions work.