1973 IMO Problems/Problem 5

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

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1973 IMO Problems/Problem 5

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