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BMO Shortlist 2021 C2

Source: BMO Shortlist 2021

May 8, 2022
Balkanshortlist2021combinatoricsDivisibilitymaximum

Problem Statement

Let KK and N>KN > K be fixed positive integers. Let nn be a positive integer and let a1,a2,...,ana_1, a_2, ..., a_n be distinct integers. Suppose that whenever m1,m2,...,mnm_1, m_2, ..., m_n are integers, not all equal to 00, such that ∣mi∣≤K\mid{m_i}\mid \le K for each ii, then the sum ∑i=1nmiai\sum_{i = 1}^{n} m_ia_i is not divisible by NN. What is the largest possible value of nn?
Proposed by Ilija Jovcevski, North Macedonia
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