ILL 1992 Problem 24

(a) Show that there exists exactly one function $f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:(i) if $0 < q < \frac 12$ , then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ (ii) if $1 < q \leq 2$ , then $f(q) = 1+f(q + 1);$ (iii) $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ (b) Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$