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The South African Mathematical Olympiad 2007

Source: Problem 5

August 22, 2008
functioninductionalgebra proposedalgebra

Problem Statement

Let Z Z and R R denote the sets of integers and real numbers, respectively. Let f:Z→R f: Z \rightarrow R be a function satisfying: (i) f(n)≥0 f(n) \ge 0 for all n∈Z n \in Z (ii) f(mn)\equal{}f(m)f(n) for all m,n∈Z m,n \in Z (iii) f(m\plus{}n) \le max(f(m),f(n)) for all m,n∈Z m,n \in Z (a) Prove that f(n)≤1 f(n) \le 1 for all n∈Z n \in Z (b) Find a function f:Z→R f: Z \rightarrow R satisfying (i), (ii),(iii) and 0<f(2)<1 0<f(2)<1 and f(2007) \equal{} 1
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