International Mathematical Olympiad 1973 Problem 5

$G$ is a set of non-constant functions of the real variable $x$ of the form $$f(x) = ax + b, \text{ } a \text{ and } b \text{ are real numbers,}$$ and $G$ has the following properties:

(a) If $f$ and $g$ are in $G$, then $g \circ f$ is in $G$; here $(g \circ f)(x) = g[f(x)]$.

(b) If $f$ is in $G$, then its inverse $f^{-1}$ is in $G$; here the inverse of $f(x) = ax + b$ is $f^{-1}(x) = (x - b)/a$.

(c) For every $f$ in $G$, there exists a real number $x_f$ such that $f(x_f) = x_f$.

Prove that there exists a real number $k$ such that $f(k) = k$ for all $f$ in $G$.