IMO Shortlist 2008 problem A3

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Dodao/la: arhiva
2. travnja 2012.
Let S\subseteq\mathbb{R} be a set of real numbers. We say that a pair (f, g) of functions from S into S is a Spanish Couple on S, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. f(x) < f(y) and g(x) < g(y) for all x, y\in S with x < y;

(ii) The inequality f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right) holds for all x\in S.

Decide whether there exists a Spanish Couple on the set S = \mathbb{N} of positive integers; on the set S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}

Proposed by Hans Zantema, Netherlands
Izvor: Međunarodna matematička olimpijada, shortlist 2008