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Construct a triangle with area ≤ A (IMO SL 1987-P19)

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August 19, 2010
geometrytrigonometrygeometric inequalityTrigonometric inequalityIMO Shortlist

Problem Statement

Let α,β,γ\alpha,\beta,\gamma be positive real numbers such that α+β+γ<π\alpha+\beta+\gamma < \pi, α+β>γ\alpha+\beta > \gamma,β+γ>α \beta+\gamma > \alpha, γ+α>β.\gamma + \alpha > \beta. Prove that with the segments of lengths sin⁡α,sin⁡β,sin⁡γ\sin \alpha, \sin \beta, \sin \gamma we can construct a triangle and that its area is not greater than A=18(sin⁡2α+sin⁡2β+sin⁡2γ).A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).
Proposed by Soviet Union
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