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A nice functional divisibility problem

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Senior D2 P6

December 21, 2022
number theoryfunctionsDivisibilitypositive integers

Problem Statement

Denote by N\mathbb{N} the set of positive integers. Find all functions f:N→Nf:\mathbb{N} \rightarrow \mathbb{N} such that:
• For all positive integers a>20232023a> 2023^{2023} it holds that f(a)≤af(a) \leq a. • a2f(b)+b2f(a)f(a)+f(b)\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)} is a positive integer for all a,b∈Na,b \in \mathbb{N}.
Proposed by Nikola Velov
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