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a_{2i} is a permutation of {1,2,3...,n} iff a_1-a_2n=n

Source: French TST 2002

June 17, 2011
absolute valuecombinatorics proposedcombinatorics

Problem Statement

Let nn be a positive integer and let (a1,a2,…,a2n)(a_1,a_2,\ldots ,a_{2n}) be a permutation of 1,2,…,2n1,2,\ldots ,2n such that the numbers ∣ai+1−ai∣|a_{i+1}-a_i| are pairwise distinct for i=1,…,2n−1i=1,\ldots ,2n-1. Prove that {a2,a4,…,a2n}={1,2,…,n}\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\} if and only if a1−a2n=na_1-a_{2n}=n.
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