What an EXCELLENT problem
by v_Enhance, Jun 14, 2012, 4:54 AM
(Balkan MO) Prove that for positive 
we have
![\[ \frac{2}{a(b+c)} + \frac{2}{b(c+a)} + \frac{2}{c(a+b)} \ge \frac{27}{(a+b+c)^2}. \]](//latex.artofproblemsolving.com/8/9/2/892731ec6af67936d03a690ea10ba92f04835162.png)
we have![\[ \frac{2}{a(b+c)} + \frac{2}{b(c+a)} + \frac{2}{c(a+b)} \ge \frac{27}{(a+b+c)^2}. \]](http://latex.artofproblemsolving.com/8/9/2/892731ec6af67936d03a690ea10ba92f04835162.png)
7 Comments
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by
dragon96, Jun 14, 2012, 4:59 AM
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WLOG 
, so this is
symmetric sum at least 
.
Note that
is at least 
by Cauchy.
Also,, so 
and then we get our expression is at least 
by Cauchy.
Now, if we show that
is at most 
when , we will be done.
Note that
, and 
is at least . Thus, is at most , and our expression is at least . We are finished yay.
Smoothing obviously kills this, and in fact kills the problem as early as. However, other ways are better.
, so this is
symmetric sum at least 
.Note that
is at least 
by Cauchy.Also,
, so 
and then we get our expression is at least 
by Cauchy.Now, if we show that
is at most 
when , we will be done.Note that
, and 
is at least . Thus, is at most , and our expression is at least . We are finished yay.Smoothing obviously kills this, and in fact kills the problem as early as
. However, other ways are better.
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by
AwesomeToad, Jun 14, 2012, 2:06 PM
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and the trivial 
,
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![\[ \left(\sum \left( \frac{b+c}{2} \right)\right)\left( \sum a \right) \left( \sum \frac{2}{a(b+c)} \right) \ge 3^3 \]](http://latex.artofproblemsolving.com/e/5/0/e508f6de7fc83404efade55a729353103cad0415.png)
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by
Mewto55555, Jun 15, 2012, 5:14 AM
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