MathDB
Sine inequality

Source: RMO 2003, District Round

May 29, 2006
trigonometryinequalitiesgeometry proposedgeometry

Problem Statement

(a) If ABC\displaystyle ABC is a triangle and M\displaystyle M is a point from its plane, then prove that AMsin⁡A≤BMsin⁡B+CMsin⁡C. \displaystyle AM \sin A \leq BM \sin B + CM \sin C . (b) Let A1,B1,C1\displaystyle A_1,B_1,C_1 be points on the sides (BC),(CA),(AB)\displaystyle (BC),(CA),(AB) of the triangle ABC\displaystyle ABC, such that the angles of △A1B1C1\triangle A_1 B_1 C_1 are A1^=α,B1^=β,C1^=γ\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma. Prove that ∑AA1sin⁡α≤∑BCsin⁡α. \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . Dan Ştefan Marinescu, Viorel Cornea
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