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sum c_ix_i is not divisible by n^3 2016 Ecuador NMO (OMEC) 3.6

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September 17, 2021
number theorydivisible

Problem Statement

A positive integer nn is "olympic" if there are nn non-negative integers x1,x2,...,xnx_1, x_2, ..., x_n that satisfy that: ∙\bullet There is at least one positive integer jj: 1≤j≤n1 \le j \le n such that xj≠0x_j \ne 0. ∙\bullet For any way of choosing nn numbers c1,c2,...,cnc_1, c_2, ..., c_n from the set {−1,0,1}\{-1, 0, 1\}, where not all cic_i are equal to zero, it is true that the sum c1x1+c2x2+...+cnxnc_1x_1 + c_2x_2 +... + c_nx_n is not divisible by n3n^3. Find the largest positive "olympic" integer.
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