A hard inequality from 2015 Taiwan TST (Open Problem)

Let a,b,c,da,b,c,d be any real numbers such that a+b+c+d=0a+b+c+d=0, prove that 1296(a7+b7+c7+d7)2≤637(a2+b2+c2+d2)7.1296(a^7+b^7+c^7+d^7)^2\le 637(a^2+b^2+c^2+d^2)^7.

Remark: I don't have a solution for now, anyone would like to try and prove it?

#Algebra

Note by ChengYiin Ong
4 months 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} ai−1 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 sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

On the RHS, substituting d=−a−b−cd = -a-b-c and simplifying further results in 637[(a+b)2+(b+c)2+(c+a)2]7637[(a+b)^2+(b+c)^2+(c+a)^2]^7. Not sure if something similar can be done with the LHS.

Elijah L - 4 months ago

(a+b)⁷=a⁷+b⁷+7ab(a+b)(a²+ab+b²)² Since a+b+c+d=0 a⁷+b⁷+c⁷+d⁷=a⁷+b⁷+c⁷-(a+b+c)⁷ =7(a+b)(b+c)(c+a)((a²+b²+c²+ab+bc+ca)²-abc(a+b+c)) [Take x=b+c,y=c+a,z=a+b] =7xyz(4(x²+y²+z²)²+2(x²y²+y²z²+z²x²)-(x⁴+y⁴+z⁴))/16 =7xyz(3(x⁴+y⁴+z⁴)+10(x²y²+y²z²+z²x²))/16 As mentioned by mention[3990831:Elijah L]. above, a²+b²+c²+d²=x²+y²+z². Substituting these values back to our original inequality and simplifying, we get, 81x²y²z²(3(x⁴+y⁴+z⁴)+10(x²y²+y²z²+z²x²))²<=208(x²+y²+z²)⁷ By AM-GM inequality we have, (x²+y²+z²)³>=27x²y²z², we just need to prove, 208(x²+y²+z²)⁴>=3(3(x⁴+y⁴+z⁴)+10(x²y²+y²z²+z²x²))² For this huge second inequality, just bring all terms to lhs, and observe that coefficient of any possible term is positive if the expression is expanded(just put the expression in wolfram alpha). Thus we are done.

Kushal Dey - 1 month ago

Anyway, really nice problem. After many days i found such difficult algebra problem. Thanks for posting Cheng Yinn Ong.

Kushal Dey - 1 month ago
×

Problem Loading...

Note Loading...

Set Loading...