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Problem 2

Part of 1986 French Mathematical Olympiad

Problems(1)

inequalities on four points in a plane, existence

Source: France 1986 P2

5/19/2021
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?
inequalitiesGeometric Inequalitiesgeometry