MathDB

8

Part of 1988 IMO Shortlist

Problems(1)

Vectors with zero sum in the plane

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

10/22/2005
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}.
vectoralgebracombinatoricscombinatorial geometryInequalityIMO Shortlist