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(absin2C+bcsin2A+casin2B){ab+bc+ca)<=\sqrt3/2

Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p3 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 9, 2022
geometrytrigonometryTrigonometric inequalityinequalitiesgeometric inequality

Problem Statement

Let ABCABC be a triangle whose side lengths are opposite the angle A,B,CA,B,C are a,b,ca,b,c respectively. Prove that absin⁡2C+bcsin⁡2A+casin⁡2Bab+bc+ca≤32\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}.
(nooonuii)
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