HMMT 2007 T11

Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that

$$\begin{aligned} f(x) f(y) & =f(x)+f(y)-f(x y) \\ 1+f(x+y) & =f(x y)+f(x) f(y) \end{aligned}$$

for all rational numbers $x, y$.