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Largest n with a strange condition

Source: Romania JBMO TST 2022 Day 2 Problem 2

May 15, 2022
number theoryRomanian TSTJBMO TST

Problem Statement

Find the largest positive integer nn such that the following is true: There exists nn distinct positive integers x1, x2,…,xnx_1,~x_2,\dots,x_n such that whatever the numbers a1, a2,…,an∈{−1,0,1}a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\} are, not all null, the number n3n^3 do not divide ∑k=1nakxk\sum_{k=1}^n a_kx_k.
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