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
5 months, 2 weeks ago

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

A Brilliant Member - 5 months, 1 week 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 - 5 months, 1 week 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.

A Brilliant Member - 5 months, 1 week ago

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@A Brilliant Member Ok. I was wondering if it could be solved using inequalities/theorems like these.

Hua Zhi Vee - 5 months, 1 week 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 - 5 months, 1 week ago

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Wow... Why did you decide to solve it then?) Great job. Earlier I like to solve some math and logic tests... I was good at this, but I could make money with it... Then I read that mobile app optimization reviews is a good source to make money, because it is connected with mobile industry. Don't regreat about my decision.

Bob Flattern - 4 months, 2 weeks ago

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RNA samples offered by Creative Bioarray isolates from human normal, fetal, diseased, and tumor tissues, mouse, rat, monkey, and plant tissues that might difficult to obtain because of anatomical complexity, size or high RNase tissue content.

Wendy Wilson - 3 weeks, 1 day ago

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