Inequalities again

by utkarshgupta, Nov 8, 2015, 5:21 AM

I guess I am back to inequalities

Problem (China TST 2005) :
Let $a$ , $b$ , $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$ . Prove that
$\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3$

Soution :

The inequality can be rewritten as
$\sum_{a,b,c}\frac{a^2-bc}{a^2-bc+1} \ge 0$ $\iff \sum_{a,b,c} (a^2-bc)(b^2ca+1)(c^2-ab+1) \ge 0$ $\iff \sum_{a,b,c} (a^2-bc)(b^2c^2-b^3a+b^2-c^3a+a^2bc-ca+c^2 -ab) \ge 0$ $\iff \sum_{a,b,c} (-a^3b^3-a^3c^3-b^3c^3 + a^2b^2+a^2c^2 +abc(a^2+b^2+c^3)-a^3b-a^3c-b^3c-bc^3 +a^2 -bc +abc(b+c)) \ge 0$
Setting $a+b+c = u$ , $v=ab+bc+ca=\frac{1}{3}$ , $w=abc$
It can be easily seen $a^3b^3+b^3c^3+c^3a^3 = v(v^2-3uw)+3w^2$
and $\sum_{a,b,c}(a^3b+a^3c+b^3c+bc^3) = 2(u^2v-2v^2-uw)$
Replacing this and other values, the inequality reduces to
$\iff -3(v(v^2-3uw)+3w^2)+2(v^2-2uw)+3w(u^3-3uv+3w) - 2(u^2v-2v^2-uw) + u^2 - 2v -v +2uw \ge 0$ $\iff -3v^3+6v^2+3u^3w-2u^2v+u^2-3v \ge 0$
Replacing $v= \frac{1}{3}$

$\iff 3u^2+27u^3w \ge 4$

Now there are 2 cases.

Case 1 : $a,b,c$ form sides of a triangle
Then using Schur's inequality,
$a^3+b^3+c^3 + 3abc \ge ab(a+b) + bc(b+c) + ca(c+a)$
$\implies u^3+9w \ge uv$
$\iff 27w \ge 4u-3u^3$

$3u^3 + 27u^3w - 4 \ge 3u^2 + u^3(4u-3u^3) - 4 = (3u^2 - 4)(1-u^4)$
Since $a,b,c$ form sides of a triangle,
$3u^2 - 4 \le 0$ and obviously $1-u^4 \le 0$

$\implies 3u^3 + 27u^3w \ge 4$ QED

Case 2 : $a,b,c$ don't form the sides of a triangle.

Normalizing, we are left to prove
$\iff u^2v^2 + u^3w \ge 4v^3$ Since $a,b,c$ are not the sides of a triangle,
Without loss of generality,
$a \ge b+c \ge \frac{u}{2}$ $u^3a \ge \frac{u^4}{2} \ge 4.5 v^2 \ge 4v^2$ $\implies u^3w \ge v^24bc \ge v^2(4bc - (b+c-a)^2)$ $\implies u^2v^2 + u^3w \ge 4v^3$
QED

This post has been edited 1 time. Last edited by utkarshgupta, Nov 8, 2015, 5:43 AM

Inequality

Conditional Inequality

Bashy

China

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Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
by rachitgoel, Sep 15, 2014, 4:21 PM
Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM

48 shouts

Elementary

Inequalities again

by utkarshgupta, Nov 8, 2015, 5:21 AM

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