IMO 2015 Shortlist N6
Let Zą0 denote the set of positive integers. Consider a function f : Zą0 Ñ Zą0. For any m,n P Zą0 we write fn pmq “ fpfp...
Category: Number Theory
Problem
Let Zą0 denote the set of positive integers. Consider a function f : Zą0 Ñ Zą0. For any m,n P Zą0 we write fn pmq “ fpfp...f looomooon n pmq...qq. Suppose that f has the following two properties: piq If m,n P Zą0, then fn pmq ´ m n P Zą0; piiq The set Zą0 z tfpnq|n P Zą0u is finite. Prove that the sequence fp1q ´ 1,fp2q ´ 2,fp3q ´ 3,... is periodic. (Singapore)