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(1/x(s-x)^2 +1/y(s-y)^2 +1/z(s-z)^ 2)>=1/2(1/(s-x)+1/(s-y)+1/(s-z)

Source: Mathcenter Contest / Oly - Thai Forum 2011 (R1) p6 sl-8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 14, 2022
geometric inequalityinequalities

Problem Statement

Let x,y,zx,y,z represent the side lengths of any triangle, and s=x+y+z2s=\dfrac{x+y+z}{2} and the area of this triangle be s\sqrt{s} square units. Prove that s(1x(s−x)2+1y(s−y)2+1z(s−z)2)≥12(1s−x+1s−y+1s−z)s\Big(\frac{1}{x(s-x)^2}+\frac{1}{y(s-y)^2}+\frac{1}{z(s-z)^ 2} \Big)\ge \frac{1}{2} \Big(\frac{1}{s-x}+\frac{1}{s-y}+\frac{1}{s-z}\Big) (Zhuge Liang)
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