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On bounding the sum of pairwise non-divisible positive integers

Source: 2022 China TST, Test 2, P3

March 28, 2022
inequalitiesnumber theoryDivisibility

Problem Statement

Let a1,a2,…,ana_1, a_2, \ldots, a_n be nn positive integers that are not divisible by each other, i.e. for any i≠ji \neq j, aia_i is not divisible by aja_j. Show that a1+a2+⋯+an≥1.1n2−2n. a_1+a_2+\cdots+a_n \ge 1.1n^2-2n.
Note: A proof of the inequality when nn is sufficient large will be awarded points depending on your results.
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