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New (easy )
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tranthanhnam
Riemann Hypothesis
Riemann Hypothesis


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#1
New (easy )
leta,b,c >0 and abc=1.
prove that :
a^2+b^2+c^2 +3 \geq 2(ab+bc+ca)
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tinh yeu cha la gi ca , va dieu dai dot nhat la ket hon chi vi tinh yeu

PostPosted: Sat Feb 25, 2006 5:17 pm  Back to top 
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Nameless
Poincare Conjecture
Poincare Conjecture

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#2
Re: new (easy )
tranthanhnam wrote:
leta,b,c >0 and abc=1.
prove that :
a^2+b^2+c^2 +3 \geq 2(ab+bc+ca)

Too easy. Just use Schur.
\sum a^2+3\sqrt[3]{a^2b^2c^2}\ge \sum(\sqrt[3]{a^4b^2}+\sqrt[3]{a^4c^2}\geq 2(ab+bc+ca)

PostPosted: Sat Feb 25, 2006 6:08 pm  Back to top 
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darij grinberg
Birch & Swinnerton Dyer
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#3
Easy: yes; new: nope
See also

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=18430
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=19076
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=50745

and a more general version in

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=49049
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=19666 (beginning with post #2)
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=40979 (beginning with post #3)

Darij
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Now the die is cast, the first step taken, a glimmer of hope lights up our lives
Visions of the past, dreams forsaken forming right under our eyes
We are alive...

PostPosted: Sun Feb 26, 2006 2:06 am  Back to top 
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Davron
Yang-Mills Theory
Yang-Mills Theory

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#4
Darij Grinberg

I have some questions for you: why dont you post a problem but
everytime solve them send the problems you had enjoyed ...
Please ...
- one more question as i remember before the exam in the IMO 2005 you and your teamates were saying womething in your lanquage with a loud voice can you say what were you saying please ...

Davron

PostPosted: Sun Feb 26, 2006 2:31 am  Back to top 
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Ji Chen
Yang-Mills Theory
Yang-Mills Theory

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#5
tranthanhnam wrote:
Let a,b,c > 0 and abc = 1, prove that

a^2 + b^2 + c^2 + 3 \geq 2( bc + ca + ab).
Mr. Qing Song pointed out that this problem is Example 1 in :

Yu-Zhong Weng, Structuring quadratic function to prove inequalities (Chinese), High School Science References (Nanning), 1998, No. 6, page 14. (ISSN: 1002-6363)

By the way,

the polynomial 2(1 - bc - ca - ab) + a^2 + b^2 + c^2 + a^2b^2c^2 is nonnegetive for real numbers a,b,c,

but cannot be represented as a sum of squares of other polynomials with real coefficients.

See also : http://www.artofproblemsolving.com/Forum/viewtopic.php?t=24093

PostPosted: Thu Nov 05, 2009 8:10 pm  Back to top 
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