Olympiad Inequality - A not-so-Open Problem

I saw this problem online some time ago, and I have been trying to solve this inequality:

\(x,y,z >0\), prove \[\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}\]

This question was asked online years ago and no one has proved it with an "elegant" way. I decided to share it here to all who haven't seen this problem yet, as I find it interesting and exciting while solving the problem (although I haven't found an elegant proof) XD

Have fun!

P.S. I will get "badges" if many clicks into the link "online" above :)


Upon typing the title I remembered a quote I saw in Evan Chen's book :D :

Graders received some elegant solutions, some not-so-elegant solutions, and some so-not-elegant solutions. --MOP 2012

Note by Hua Zhi Vee
1 week, 5 days ago

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Muito obrigado pela solução detalhada para esta tarefa. é incrivelmente interessante. Eu gentilmente usá-lo em hashing24 é confiavel Isso me ajuda a encontrar algum tipo de interconexão entre indicadores

Mari Klark - 1 week, 3 days ago

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There is a solution that involves using symmetric polynomials, but you will need a good computer to do this.

Gennady Notowidigdo - 1 week, 4 days ago

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This problem can be solved using just pen and paper.

This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.

Hua Zhi Vee - 1 week, 4 days ago

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Of course you can, but a proper proof would require symmetric polynomials, for which simplifying them (in this case) would be a right piece of work for a computer, let alone a human being. Fortunately, I have a very good PC that can help me out here; many others on this website would struggle to do so.

Gennady Notowidigdo - 1 week, 4 days ago

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@Gennady Notowidigdo Ok. I was wondering if it could be solved using inequalities/theorems like these.

Hua Zhi Vee - 1 week, 2 days ago

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