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f(q)= 1 + f(q/(1-2q))

Source: Israel Grosman Memorial Mathematical Olympiad 1995 p6

February 15, 2020
functional equationfunctionalalgebra

Problem Statement

(a) Prove that there is a unique function f:Q→Qf : Q \to Q satisfying: (i) f(q)=1+f(q1−2q)f(q)= 1 + f\left(\frac{q}{1-2q}\right) for 0<q<120<q< \frac12 (ii) f(q)=1+f(q−1)f(q)= 1 + f(q-1) for 1<q≤21<q\le 2 (iii) f(q)f(1q)=1f(q)f\left(\frac{1}{q}\right)=1 for all q∈Q+q\in Q^+ (b) For this function ff , find all r∈Q+r\in Q^+ such that f(r)=rf(r) = r
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