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Sum of the lengths of the projections of the given segments

Source: IMO Longlist 1989, Problem 73

September 18, 2008
symmetrygeometryperimetercalculustrigonometryintegrationinequalities

Problem Statement

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line l l such that the sum of the lengths of the projections of the given segments to the line l l is less than 2π. \frac{2}{\pi}.
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