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[a,b] contains infinitely many 2020-good numbers, real x= sum 1/a_i, a_i in N

Source: 2020 Czech-Polish-Slovak Match p2

October 8, 2020
combinatoricsalgebra

Problem Statement

Given a positive integer nn, we say that a real number xx is nn-good if there exist nn positive integers a1,...,ana_1,...,a_n such that x=1a1+...+1an.x=\frac{1}{a_1}+...+\frac{1}{a_n}. Find all positive integers kk for which the following assertion is true: if a,ba,b are real numbers such that the closed interval [a,b][a,b] contains infinitely many 20202020-good numbers, then the interval [a,b][a,b] contains at least one kk-good number.
(Josef Tkadlec, Czech Republic)
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