IMO 1978 LL CUB5

Prove that for any triangle ABC there exists a point P in the

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IMO 1978 LL CUB5

Origin: CUB

Problem

Prove that for any triangle ABC there exists a point P in the plane of the triangle and three points A′, B′, and C′ on the lines BC, AC, and AB respectively such that AB \cdot PC′ = AC \cdot PB′ = BC \cdot PA′ = 0.3M 2, where M = max{AB, AC, BC}.