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Number theory sequences : sums divisible by n

Source: IMO Shortlist 2017 N3

July 10, 2018
IMO Shortlistnumber theory

Problem Statement

Determine all integers n≥2 n\geq 2 having the following property: for any integers a1,a2,…,ana_1,a_2,\ldots, a_n whose sum is not divisible by nn, there exists an index 1≤i≤n1 \leq i \leq n such that none of the numbers ai,ai+ai+1,…,ai+ai+1+…+ai+n−1a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1} is divisible by nn. Here, we let ai=ai−na_i=a_{i-n} when i>ni >n.
Proposed by Warut Suksompong, Thailand
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