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A number theory problem from Iran TST

Source: Iranian TST 2018, third exam day 2, problem 4

April 19, 2018
number theoryIranIranian TST

Problem Statement

We say distinct positive integers  a1,a2,…,ana_1,a_2,\ldots ,a_n are "good" if their sum is equal to the sum of all pairwise gcd⁡\gcd 's among them. Prove that there are infinitely many nn s such that nn good numbers exist.
Proposed by Morteza Saghafian
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