I was trying to prove an inequality with 3 variables, and after simplification, it boiled down to trying to prove
$x^2 y+xz^2 +y^2 z \geq x+y+z$, where $xyz=1$, and $x,y,z$ are all positive real numbers.
Once, I prove this I would be able to prove the overall inequality, however I am stuck at this part. The original question is supposed to be an olympiad style question, so i hope that i can prove this lemma without Lagrange Multipliers, etc.
Sincere thanks for any help!
(I have tried AM-GM, but that would lead to $x^2 y+xz^2 +y^2 z \geq 3\sqrt[3]{x^3y^3z^3}=3$ which is not sharp enough)