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exist vectors with sum <= \sqrt2/8, among n vectors with sum <=1

Source: Czech and Slovak Olympiad 1985, National Round, Problem 3

September 11, 2024
vectorinequalitiescombinatoricscombinatorial geometrygeometry

Problem Statement

If u1→,u2→,...,un→\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n} be vectors in the plane such that the sum of their lengths is at least 11, then between them we find vectors whose sum is a vector of length at least 2/8\sqrt2/8. Prove it.
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