IMO Shortlist 2001 problem G4


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2. travnja 2012.
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Let M be a point in the interior of triangle ABC. Let A' lie on BC with MA' perpendicular to BC. Define B' on CA and C' on AB similarly. Define

p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.

Determine, with proof, the location of M such that p(M) is maximal. Let \mu(ABC) denote this maximum value. For which triangles ABC is the value of \mu(ABC) maximal?
Izvor: Međunarodna matematička olimpijada, shortlist 2001