IMO Shortlist 1973 problem 17

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Dodao/la: arhiva
April 2, 2012
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.
Source: Međunarodna matematička olimpijada, shortlist 1973