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Let ABC be a triangle. Prove that there exists a point D on the side AB of the triangle ABC, such that CD is the geometric mean of AD and DB, iff the triangle ABC satisfies the inequality \sin A\sin B\le\sin^2\frac{C}{2}.

CommentAlternative formulation, from IMO ShortList 1974, Finland 2: We consider a triangle ABC. Prove that: \sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right) is a necessary and sufficient condition for the existence of a point D on the segment AB so that CD is the geometrical mean of AD and BD.

Slični zadaci

#NaslovOznakeRj.KvalitetaTežina
1734IMO Shortlist 1988 problem 270
1314IMO Shortlist 1968 problem 90
1250IMO Shortlist 1967 problem 40
1150IMO Shortlist 1960 problem 41
1144IMO Shortlist 1959 problem 47
1142IMO Shortlist 1959 problem 221