[tex]x+z\geq \frac{ (\sqrt{x}+\sqrt{z})^2}{2} ; y+z\geq \frac{ (\sqrt{x}+\sqrt{z})^2}{2} => x+y+2z \geq \frac{ (\sqrt{x}+\sqrt{z})^2}{2}+\frac{ (\sqrt{x}+\sqrt{z})^2}{2} \geq \frac{(\sqrt{x}+\sqrt{y}+2\sqrt{z})^2}{4} => \sqrt{\frac{xy}{x+y+2z}}\leq \sqrt{\frac{4xy}{(\sqrt{x}+\sqrt{y}+2\sqrt{z})^2}}[/tex]
Có [tex]\frac{1}{a+b}\leq \frac{1}{4}(\frac{1}{a}+\frac{1}{b}) => \frac{2\sqrt{xy}}{\sqrt{x}+\sqrt{z}+\sqrt{y}+\sqrt{z}}\leq \frac{\sqrt{xy}}{2}(\frac{1}{\sqrt{x}+\sqrt{z}}+\frac{1}{\sqrt{y}+\sqrt{z}})[/tex]
Tương tự với 2 cái kia rồi cộng vào là ra đpcm