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Prove this hard inequality on triangles

Source: 2009 Jozsef Wildt International Mathematical Competition

April 26, 2020
trigonometryinequalities

Problem Statement

Let a triangle △ABC\triangle ABC and the real numbers xx, yy, z>0z>0. Prove that xncos⁡A2+yncos⁡B2+zncos⁡C2≥(yz)n2sin⁡A+(zx)n2sin⁡B+(xy)n2sin⁡Cx^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C
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