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inequalities on four points in a plane, existence

Source: France 1986 P2

May 19, 2021
inequalitiesGeometric Inequalitiesgeometry

Problem Statement

Points A,B,CA,B,C, and MM are given in the plane. (a) Let DD be the point in the plane such that DA≤CADA\le CA and DB≤CBDB\le CB. Prove that there exists point NN satisfying NA≤MA,NB≤MBNA\le MA,NB\le MB, and ND≤MCND\le MC. (b) Let A′,B′,C′A',B',C' be the points in the plane such that A′B′≤AB,A′C′≤AC,B′C′≤BCA'B'\le AB,A'C'\le AC,B'C'\le BC. Does there exist a point M′M' which satisfies the inequalities M′A′≤MA,M′B′≤MB,M′C′≤MCM'A'\le MA,M'B'\le MB,M'C'\le MC?
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