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Function defined on positive rationals

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September 1, 2010
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Problem Statement

(a) Show that there exists exactly one function f:Q+→Q+ f : \mathbb Q^+ \to \mathbb Q^+ satisfying the following conditions: (i) if 0<q<120 < q < \frac 12, then f(q)=1+f(q1−2q);f(q)=1+f \left( \frac{q}{1-2q} \right);
(ii) if 1<q≤21 < q \leq 2, then f(q)=1+f(q+1);f(q) = 1+f(q + 1);
(iii) f(q)f(1/q)=1f(q)f(1/q) = 1 for all q∈Q+.q \in \mathbb Q^+.
(b) Find the smallest rational number q∈Q+q \in \mathbb Q^+ such that f(q)=1992.f(q) = \frac{19}{92}.
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