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p_i +p_{i+1} +...+p_j is divisible by n

Source: 2019 RMM Shortlist C3

June 19, 2020
combinatoricsdivisiblepermutation

Problem Statement

Fix an odd integer n>1n > 1. For a permutation pp of the set {1,2,...,n}\{1,2,...,n\}, let S be the number of pairs of indices (i,j)(i, j), 1≤i≤j≤n1 \le i \le j \le n, for which pi+pi+1+...+pjp_i +p_{i+1} +...+p_j is divisible by nn. Determine the maximum possible value of SS.
Croatia
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