BALTIC-WAY 1992 Problem 11

Let $\mathbb{Q}^{+}$denote the set of positive rational numbers. Show that there exists one and only one function $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$satisfying the following conditions:

(i) If $0<q<\frac{1}{2}$ then $f(q)=1+f\left(\frac{q}{1-2 q}\right)$.

(ii) If $1<q \leq 2$ then $f(q)=1+f(q-1)$.

(iii) $f(q) \cdot f\left(\frac{1}{q}\right)=1$ for all $q \in \mathbb{Q}^{+}$.