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Vectors with zero sum in the plane

Source: IMO ShortList 1988, Problem 8, France 3, Problem 11 of ILL

October 22, 2005
vectoralgebracombinatoricscombinatorial geometryInequalityIMO Shortlist

Problem Statement

Let u1,u2,…,um u_1, u_2, \ldots, u_m be m m vectors in the plane, each of length ≤1, \leq 1, with zero sum. Show that one can arrange u1,u2,…,um u_1, u_2, \ldots, u_m as a sequence v1,v2,…,vm v_1, v_2, \ldots, v_m such that each partial sum v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m has length less than or equal to 5. \sqrt {5}.
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