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IMO ShortList 2001, geometry problem 4

Source: IMO ShortList 2001, geometry problem 4

September 30, 2004
geometrytrigonometrymaximizationTriangleperpendicularIMO Shortlist

Problem Statement

Let MM be a point in the interior of triangle ABCABC. Let A′A' lie on BCBC with MA′MA' perpendicular to BCBC. Define B′B' on CACA and C′C' on ABAB similarly. Define p(M)=MA′⋅MB′⋅MC′MA⋅MB⋅MC. p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. Determine, with proof, the location of MM such that p(M)p(M) is maximal. Let μ(ABC)\mu(ABC) denote this maximum value. For which triangles ABCABC is the value of μ(ABC)\mu(ABC) maximal?
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