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Problem 2, BMO 2020

Source: Problem 2, BMO 2020

November 1, 2020
algebraBMO

Problem Statement

Denote Z>0={1,2,3,...}\mathbb{Z}_{>0}=\{1,2,3,...\} the set of all positive integers. Determine all functions f:Z>0→Z>0f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0} such that, for each positive integer nn, i)∑k=1nf(k)\hspace{1cm}i) \sum_{k=1}^{n}f(k) is a perfect square, and \vspace{0.1cm} ii)f(n)\hspace{1cm}ii) f(n) divides n3n^3.
Proposed by Dorlir Ahmeti, Albania
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