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n sums are positive

Source: Thailand TSTST 2021, test 3, P3

August 16, 2022
combinatorics

Problem Statement

An odd positive integer nn is called pretty if there exists at least one permutation a1,a2,...,ana_1, a_2,..., a_n, of 1,2,...,n1,2,...,n, such that all nn sums a1−a2+a3−...+ana_1-a_2+a_3-...+a_n, a2−a3+a4−...+a1a_2-a_3+a_4-...+a_1,..., an−a1+a2−...+an−1a_n-a_1+a_2-...+a_{n-1} are positive. Find all pretty integers.
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