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Inequality with permutations

Source: 2019 Belarus Team Selection Test 7.3

September 2, 2019
inequalitiesalgebra

Problem Statement

Given a positive integer nn, determine the maximal constant CnC_n satisfying the following condition: for any partition of the set {1,2,…,2n}\{1,2,\ldots,2n \} into two nn-element subsets AA and BB, there exist labellings a1,a2,…,ana_1,a_2,\ldots,a_n and b1,b2,…,bnb_1,b_2,\ldots,b_n of AA and BB, respectively, such that (a1−b1)2+(a2−b2)2+…+(an−bn)2≥Cn. (a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
(B. Serankou, M. Karpuk)
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