IMO 2008 Shortlist A3
Let S ⊆ 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 the...
Category: Algebra
Problem
Let S ⊆ 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 ∈ S with x < y; (ii) The inequality f(g(g(x))) < g(f(x)) holds for all x ∈ S. Decide whether there exists a Spanish Couple (a) on the set S = N of positive integers; (b) on the set S = {a − 1/b : a,b ∈ N}.