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F^n(0)=0 iff f^n(t)=t

Source: Problem 5, Brazilian MO, 1990

March 19, 2006
functionquadraticsalgebradomaininductionlinear algebramatrix

Problem Statement

Let f(x)=ax+bcx+df(x)=\frac{ax+b}{cx+d} Fn(x)=f(f(f...f(x)...))F_n(x)=f(f(f...f(x)...)) (with n f′sn\ f's)
Suppose that f(0)≠0f(0) \not =0, f(f(0))≠0f(f(0)) \not = 0, and for some nn we have Fn(0)=0F_n(0)=0, show that Fn(x)=xF_n(x)=x (for any valid x).
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