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Infinite positive integer sequence with an eventual cubic lower bound

Source: 2019 Philippine IMO TST1 Problem 3

May 4, 2022
number theorydivisorBoundinginfinite sequence

Problem Statement

Let a1,a2,a3,…a_1, a_2, a_3,\ldots be an infinite sequence of positive integers such that a2≠2a1a_2 \ne 2a_1, and for all positive integers mm and nn, the sum m+nm + n is a divisor of am+ana_m + a_n. Prove that there exists an integer MM such that for all n>Mn > M, we have an≥n3a_n \ge n^3.
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