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IMO ShortList 1988, Geometric Inequality

Source: IMO ShortList 1988, Problem 12, Greece 2, Problem 22 of ILL

October 22, 2005
inequalitiesgeometryratioTrianglearea of a triangleIMO Shortlist

Problem Statement

In a triangle ABC, ABC, choose any points K∈BC,L∈AC,M∈AB,N∈LM,R∈MK K \in BC, L \in AC, M \in AB, N \in LM, R \in MK and F∈KL. F \in KL. If E1,E2,E3,E4,E5,E6 E_1, E_2, E_3, E_4, E_5, E_6 and E E denote the areas of the triangles AMR,CKR,BKF,ALF,BNM,CLN AMR, CKR, BKF, ALF, BNM, CLN and ABC ABC respectively, show that E≥8⋅E1E2E3E4E5E66. E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
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