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\Sum a_1=0 from i=1,...,2n, among all pairs 2n-1 have sum >=0

Source: Danube junior 2014 p4

July 22, 2019
algebracombinatoricsSum

Problem Statement

Consider the real numbers a1,a2,...,a2na_1,a_2,...,a_{2n} whose sum is equal to 00. Prove that among pairs (ai,aj),i<j(a_i,a_j) , i<j where i,j∈{1,2,...,2n} i,j \in \{1,2,...,2n\} .there are at least 2n−12n-1 pairs with the property that ai+aj≥0a_i+a_j\ge 0.
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