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Prove that the sequence is constant

Source: KJMO 2019 p4

January 8, 2021
SequenceinequalitiesKJMOconstant

Problem Statement

{an}\{a_{n}\} is a sequence of natural numbers satisfying the following inequality for all natural number nn: (a1+⋯+an)(1a1+⋯+1an)≤n2+2019(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019 Prove that {an}\{a_{n}\} is constant.
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