IMO 1973 Problem 5

$G$ is a set of non-constant functions $f$ . Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$ . If $f$ and $g$ are in $G$ , then so is $fg$ , where $fg$ is defined by $fg(x)=f(g(x))$ . If $f$ is in $G$ , then so is the inverse $f^{-1}$ . If $f(x)=ax+b$ , then $f^{-1}(x)= \frac{x-b}{a}$ . Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$ . Prove that all the functions in $G$ have a common fixed point.