IMO 1992 LL ICE24

Origin: ICE

Problem

Let Q+ denote the set of nonnegative rational numbers. Show that there exists exactly one function f : Q+ \toQ+ satisfying the follow- ing conditions: (i) if 0 < q < 1 2, then f(q) = 1 + f  q 1−2q ; (ii) if 1 < q \leq2, then f(q) = 1 + f(q + 1); (iii) f(q)f(1/q) = 1 for all q \inQ+. Find the smallest rational number q \inQ+ such that f(q) = 19/92.