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>0x,y,z >0, prove x48x3+5y3+y48y3+5z3+z48z3+5x3x+y+z13\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
11 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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

A Former Brilliant Member - 11 months, 1 week ago

Log in to reply

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 - 11 months, 1 week ago

Log in to reply

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 Former Brilliant Member - 11 months, 1 week ago

Log in to reply

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

Hua Zhi Vee - 11 months, 1 week ago

Log in to reply

Colorectal cancer (CRC) is one of the most common and deadly cancers in the world. CRC occurs due to the accumulation of genetic interactions and alterations. At present, the molecular research progress of CRC is still slow. It has been reported that there is a close relationship between GGCT expression and CRC cells.

https://www.creative-biogene.com/genesearch/C7orf24.html

Wendy Wilson - 1 week, 5 days ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...