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\sum_{j=1}^{n}|\sum_{k=1}^{n} a_{j,p}(k)| \ge \frac{n}{2}

Source: 1984 Polish MO Finals p2

February 25, 2020
inequalitiesalgebra

Problem Statement

Let nn be a positive integer. For all i,j∈{1,2,...,n}i, j \in \{1,2,...,n\} define aj,i=1a_{j,i} = 1 if j=ij = i and aj,i=0a_{j,i} = 0 otherwise. Also, for i=n+1,...,2ni = n+1,...,2n and j=1,...,nj = 1,...,n define aj,i=−1na_{j,i} = -\frac{1}{n}. Prove that for any permutation pp of the set {1,2,...,2n}\{1,2,...,2n\} the following inequality holds: ∑j=1n∣∑k=1naj,p(k)∣≥n2\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}
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