Waste less time on Facebook — follow Brilliant.
×

Daniel Liu's Hard Inequality

Hello every body.Sorry for too late.But this PROBLEM is very hard. In 1989,Jack Garfunkel recommended a inequality like this problem to Crux magazine in Canada.It's a hard problem and didn't have any solution at that time.

If we given \(a,b,c\) are positive reals,the equality holds when \(a=b=c\) and at that time,\(k=\frac{\sqrt{3}}{\sqrt{2}}\)

But \(k=\frac{5}{4}\) must be better but we must let \(a,b,c\) are non-negative and the equality holds when \(a=b=3,c=0\) or it's permutation.

At present,there have many way to prove Jack Garfunkel's inequality. By applying full incremental variable,we have: \[a+b+c+\frac{(a-b)(b-c)(a-c)(a+b+c)}{abc}\leq \frac{5\sqrt{2}}{4}\sqrt{x^2+y^2+z^2}\]

Let \(f(t)=\frac{(x+y+z-3t)^2}{(x-t)(y-t)(z-t)}\) is decreasing and \(g(t)=\frac{5\sqrt{2}(x+y+z-3t)\sqrt{(x-t)^2+(y-t)^2+(z-t)^2}}{4}-(x+y+z-3t)^2\) is increasing in \([0;4]\). It's true because if we let \(m=x-t;n=y-t,p=z-t\),\(m,n,p > 0\) and \(A=\sqrt{m^2+n^2+p^2};B=m+n+p\),we get: \(m^2n^2p^2f'(t) \geq 0\) and \(g'(t) \leq 0 \). So that we get: \[\frac{a}{\sqrt{a+b}}+\sqrt{b}\leq \frac{5}{4}\sqrt{a+b}\] This it's right because \(RHS-LHS \geq 0\). The problem is solved. Another way is more interested:By C-S Inequality \[LHS^2\leq \sum a(5a+b+9c)\sum \frac{a}{(a+b)(5a+b+9c)}\] We need to prove: \[(a+b+c)\sum \frac{a}{(a+b)(5a+b+9c)}\leq \frac{5}{16}\] This inequality right because: \[(RHS-LHS)(16\prod (5a+b+9c))=\sum ab(a+b)(a+9b)(a-3b)^2+243\sum a^3b^2c+835\sum a^3bc^2+232\sum a^4bc+1230a^2b^2c^2\geq 0\]

The problem is solved.

Note by Ms Ht
1 year, 8 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Can you explain this clearer?

  1. What are you trying to describe?
  2. What is the process to solve such a problem
  3. How does someone come up with the various inequalities?

Calvin Lin Staff - 1 year, 7 months ago

Log in to reply

Prove this Consider \(a,b,c>0, abc=1\). Prove that

\[\sum_{\mathrm{cyc}} \frac{a^{2}+b^{2}}{a^{8}+b^{8}}\leq a^{3}+b^{3}+c^{3}\]

Shivam Jadhav - 1 year, 8 months ago

Log in to reply

Log in to reply

Ah yes, this was the SOS method you mentioned on the solutions page right? I believe it is from a China TST, as well.

Ameya Daigavane - 1 year, 8 months ago

Log in to reply

Also Vietnam TST

Ms Ht - 1 year, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...