IMO LONGLISTED PROBLEM 1970

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MATHPRITOM
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IMO LONGLISTED PROBLEM 1970

Unread post by MATHPRITOM » Wed Apr 27, 2011 7:17 pm

$\frac{bc}{b+c} $+$\frac{ca}{c+a}$+$\frac{ab}{a+b}$ $\ge$ $\frac{a+b+c}{2}$.where a,b,c>0.

MATHPRITOM
Posts: 190
Joined: Sat Apr 23, 2011 8:55 am
Location: Khulna

Re: IMO LONGLISTED PROBLEM 1970

Unread post by MATHPRITOM » Wed Apr 27, 2011 7:18 pm

a very nice problem.

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Masum
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Re: IMO LONGLISTED PROBLEM 1970

Unread post by Masum » Sat Apr 30, 2011 3:44 pm

Let, $a=\frac{1}{x},b=\frac{1}{y},c=\frac{1}{z}$
Then the inequality is reduced to \[\sum \frac{1}{x+y}\ge \frac{xy+yz+zx}{xyz}=\sum \frac{1}{2z}\]
This is true by re-arrangement inequality.
One one thing is neutral in the universe, that is $0$.

MATHPRITOM
Posts: 190
Joined: Sat Apr 23, 2011 8:55 am
Location: Khulna

Re: IMO LONGLISTED PROBLEM 1970

Unread post by MATHPRITOM » Sun May 01, 2011 1:14 pm

thanks 4 a very nice,small solution.my solution is too long.

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