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Inequality for absolute values of vectors in plane

Source: Romania TST 4 2009, Problem 2

May 4, 2012
inequalitiesvectorgeometry proposedgeometry

Problem Statement

Let m<nm<n be two positive integers, let II and JJ be two index sets such that ∣I∣=∣J∣=n|I|=|J|=n and ∣I∩J∣=m|I\cap J|=m, and let uku_k, k∈I∪Jk\in I\cup J be a collection of vectors in the Euclidean plane such that ∣∑i∈Iui∣=1=∣∑j∈Juj∣.|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|. Prove that ∑k∈I∪J∣uk∣2≥2m+n\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n} and find the cases of equality.
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