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Inequality involving vectors in the plane

Source: Romania TST 6 2010, Problem 3

August 25, 2012
inequalitiesvectoralgebra proposedalgebra

Problem Statement

Let nn be a positive integer number. If SS is a finite set of vectors in the plane, let N(S)N(S) denote the number of two-element subsets {v,v′}\{\mathbf{v}, \mathbf{v'}\} of SS such that 4 (v⋅v′)+(∣v∣2−1)(∣v′∣2−1)<0.4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. Determine the maximum of N(S)N(S) when SS runs through all nn-element sets of vectors in the plane.
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