n house in village - Iran NMO 2003 (Second Round) - Problem2
Source:
October 4, 2010
functiongeometryinequalitiesparallelogramtriangle inequalitycombinatorics proposedcombinatorics
Problem Statement
In a village, there are houses with and all of them are not collinear. We want to generate a water resource in the village. For doing this, point is better than point if the sum of the distances from point to the houses is less than the sum of the distances from point to the houses. We call a point ideal if there doesn’t exist any better point than it. Prove that there exist at most ideal point to generate the resource.