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Regional Olympiad - FBH 2008 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2008

September 18, 2018
Sequencefunctionalgebrafunctional equation

Problem Statement

Determine is there a function a:N→Na: \mathbb{N} \rightarrow \mathbb{N} such that: i)i) a(0)=0a(0)=0 ii)ii) a(n)=n−a(a(n))a(n)=n-a(a(n)), ∀n∈\forall n \in N \mathbb{N}. If exists prove: a)a) a(k)≥a(k−1)a(k)\geq a(k-1) b)b) Does not exist positive integer kk such that a(k−1)=a(k)=a(k+1)a(k-1)=a(k)=a(k+1).
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