IMO 2006 Shortlist A3
The sequence c0,c1,...,cn,... is defined by c0 = 1, c1 = 0 and cn+2 = cn+1 + cn for n ≥ 0. Consider the set S of ordered...
Category: Algebra
Problem
The sequence c0,c1,...,cn,... is defined by c0 = 1, c1 = 0 and cn+2 = cn+1 + cn for n ≥ 0. Consider the set S of ordered pairs (x,y) for which there is a finite set J of positive integers such that x = P j∈J cj, y = P j∈J cj−1. Prove that there exist real numbers α,β and m,M with the following property: An ordered pair of nonnegative integers (x,y) satisfies the inequality m < αx + βy < M if and only if (x,y) ∈ S. N. B. A sum over the elements of the empty set is assumed to be 0. (Russia)