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Analytic Number Theory again from its pleasant side

Source: RMM Extralist 2021 N2

September 18, 2023
number theoryprimesCombinatorial Number TheoryRMM Shortlist

Problem Statement

We call a set of positive integers suitable if none of its elements is coprime to the sum of all elements of that set. Given a real number ε∈(0,1)\varepsilon \in (0,1), prove that, for all large enough positive integers NN, there exists a suitable set of size at least εN\varepsilon N, each element of which is at most NN.
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