1973 IMO Problems/Problem 5
is a set of non-constant functions of the real variable of the form and has the following properties:
(a) If and are in , then is in ; here .
(b) If is in , then its inverse is in ; here the inverse of is .
(c) For every in , there exists a real number such that .
Prove that there exists a real number such that for all in .
First, observe that for each function in , if then . This is a result of (c); for example, could not be in because it does not have a fixed point. Or if , then every point is a fixed point.
Also, for each function in , if then the fixed point of is where intersects , namely where .
Now, take and , both in . By (a), and must also both be in . By (b), must also be in . Finally, by (a),
must also be in .
Using our first observation, . Rearranging, we get . Therefore, the fixed point of equals the fixed point of . Since we made no assumptions about and , this is true for all in .