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Representing pairs of real numbers in a certain way

Source: 42nd International Tournament of Towns, Senior A-Level P2, Spring 2021

February 18, 2023
algebraTournament of Towns

Problem Statement

Does there exist a positive integer nn{} such that for any real xx{} and yy{} there exist real numbers a1,…,ana_1, \ldots , a_n satisfying x=a1+⋯+an and y=1a1+⋯+1an?x=a_1+\cdots+a_n\text{ and }y=\frac{1}{a_1}+\cdots+\frac{1}{a_n}? Artemiy Sokolov
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