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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)

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    can you post the original problem please?2015-05-26

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Hint: By AM-GM, $x^2y+x^2y+xz^2\ge 3\left(x^5y^2z^2\right)^{\frac{1}{3}}=3x.$

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    @dineshdileep: It is just a hint rather than a full answer.2012-12-17