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Given $x, y, z $ are 3 positive reals such that $xyz≥1$. Prove that $$\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0.$$ This question is so complicated. I failed many times to get the proof. Can anyone help me please? Thank you.

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up vote 4 down vote accepted

This is problem №3 from IMO 2005. Here you can find its solution.

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I thought this looked familiar. Good catch! –  Byron Schmuland Aug 8 '12 at 18:19
    
How to get the first step? –  ᴊ ᴀ s ᴏ ɴ Aug 10 '12 at 9:45
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