IMO Shortlist 1974 problem 10

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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.
Izvor: Međunarodna matematička olimpijada, shortlist 1974