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Sum a sin a/2>=p

Source: Moldova TST 2006 Test II problem 3

March 25, 2006
functioninequalitiesgeometrycircumcircletrigonometryabstract algebrainequalities proposed

Problem Statement

Let a,b,ca,b,c be sides of a triangle and pp its semiperimeter. Show that a(p−b)(p−c)bc+b(p−c)(p−a)ac+c(p−a)(p−b)ab≥pa\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p
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