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Source: EGMO 2022/2

April 9, 2022
EGMOnumber theoryfunctional equationEGMO2022

Problem Statement

Let N={1,2,3,… }\mathbb{N}=\{1, 2, 3, \dots\} be the set of all positive integers. Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that for any positive integers aa and bb, the following two conditions hold: (1) f(ab)=f(a)f(b)f(ab) = f(a)f(b), and (2) at least two of the numbers f(a)f(a), f(b)f(b), and f(a+b)f(a+b) are equal.
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