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Inequality of a non-decreasing multiplicative function

Source: Nordic MO 2010 Q1

April 21, 2013
inequalitiesfunctionmodular arithmeticnumber theoryrelatively primealgebra

Problem Statement

A function f:Z+→Z+f : \mathbb{Z}_+ \to \mathbb{Z}_+, where Z+\mathbb{Z}_+ is the set of positive integers, is non-decreasing and satisfies f(mn)=f(m)f(n)f(mn) = f(m)f(n) for all relatively prime positive integers mm and nn. Prove that f(8)f(13)≥(f(10))2f(8)f(13) \ge (f(10))^2.
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