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geometric inequality with altitudes! CMO 2015 P2

Source: Canadian mathematical olympiad 2015

April 24, 2015
geometric inequalityinequalitiesgeometry

Problem Statement

Let ABCABC be an acute-angled triangle with altitudes AD,BE,AD,BE, and CFCF. Let HH be the orthocentre, that is, the point where the altitudes meet. Prove that AB⋅AC+BC⋅CA+CA⋅CBAH⋅AD+BH⋅BE+CH⋅CF≤2.\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.
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