ISL 2008 A3

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 $f(g(g(x)))<g(f(x))$ holds for all $x \in S$. Decide whether there exists a Spanish Couple (a) on the set $S=\mathbb{N}$ of positive integers; (b) on the set $S=\{a-1 / b: a, b \in \mathbb{N}\}$.