sqing wrote: In an acute $\triangle ABC$ ,Prove that\[\frac{1}{3}(\frac{tan^2A}{tanBtanC}+\frac{tan^2B}{tanCtanA}+\frac{tan^2C}{tanAtanB})+3(\frac{1}{tanA+tanB+tanC})^{\frac{2}{3}}\ge 2\]. Since $tanAtanBtanC=tanA+tanB+tanC$ then Let $x=tanA, y=tanB, z=tanC$ and $x+y+z=xyz$ It just need to show: \[\frac{x^3+y^3+z^3}{3xyz}+\frac{\sqrt[3]{xyz}{x+y+z}\ge 2\] Which is homogenization, So we can assum $xyz=1$, then just need prove: \[\frac{x^3+y^3+z^3}{3}+\frac{3}{x+y+z}\ge 2\] Which is obviously true by AM-GM and Power mean~~

huangke3847 wrote: sqing wrote: In an acute $\triangle ABC$ ,Prove that\[\frac{1}{3}(\frac{tan^2A}{tanBtanC}+\frac{tan^2B}{tanCtanA}+\frac{tan^2C}{tanAtanB})+3(\frac{1}{tanA+tanB+tanC})^{\frac{2}{3}}\ge 2\]. Since $tanAtanBtanC=tanA+tanB+tanC$ then Let $x=tanA, y=tanB, z=tanC$ and $x+y+z=xyz$ It just need to show:$\frac{x^3+y^3+z^3}{3xyz}+\frac{3\sqrt[3]{xyz}}{x+y+z}\ge 2$ Which is homogenization, So we can assum $xyz=1$, then just need prove: \[\frac{x^3+y^3+z^3}{3}+\frac{3}{x+y+z}\ge 2\] Which is obviously true by AM-GM and Power mean~~ Very nice. Thank you very much. In an acute $\triangle ABC$ ,Prove that\[\frac{1}{3}(\frac{tan^2A}{tanBtanC}+\frac{tan^2B}{tanCtanA}+\frac{tan^2C}{tanAtanB})+3(\frac{1}{tanA+tanB+tanC})^{\frac{2}{3}} \]\[\ge\frac{2}{3}(tanA+tanB+tanC)^{\frac{2}{3}}\]. here

huangke3847 wrote: sqing wrote: In an acute $\triangle ABC$ ,Prove that\[\frac{1}{3}(\frac{tan^2A}{tanBtanC}+\frac{tan^2B}{tanCtanA}+\frac{tan^2C}{tanAtanB})+3(\frac{1}{tanA+tanB+tanC})^{\frac{2}{3}}\ge 2\]. Since $tanAtanBtanC=tanA+tanB+tanC$ then Let $x=tanA, y=tanB, z=tanC$ and $x+y+z=xyz$ It just need to show: \[\frac{x^3+y^3+z^3}{3xyz}+\frac{\sqrt[3]{xyz}{x+y+z}\ge 2\] Which is homogenization, So we can assum $xyz=1$, then just need prove: \[\frac{x^3+y^3+z^3}{3}+\frac{3}{x+y+z}\ge 2\] Which is obviously true by AM-GM and Power mean~~ Please post your solution

I think he means that: $ \frac{x^3+y^3+z^3}{3}+\frac{3}{x+y+z} \ge 2\sqrt{\frac{x^3+y^3+z^3}{x+y+z}} \ge \frac{2(x+y+z)}{3} \ge 2 $

Here's my approach https://wiki-images.artofproblemsolving.com//5/5d/Trig_in_2.png