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AOPS上的一个带绝对值的不等式

陈洪葛 posted @ 12 年前 in 不等式 , 1145 阅读

For a,b,c are real numbers such that ab+bc+ca>0 with a+b+c=1,find min
P=2|ab|+2|bc|+2|ca|+5ab+bc+ca

choisiwon,Japan)

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=539142
Solution (thanks tian275461's hint)
We will prove that
P106
Without loss of generally,we can assume that a>b>c
by AM-GM inequality
2ab+2bc8ac
which leads
P10ac+5ab+bc+ca4ac10+ab+bc+ca5
Thus,it's suffice to prove
ac10+ab+bc+ca54106=1156
Or
(ac)+2(a+c)(a+c)2+ac236
Now,Using Cauchy-Schwarz and AM-GM inequality,we have
(ac)+2(a+c)(a+c)2+ac2[(ac)2+4(a+c)4(a+c)2+4ac]=2(a+c)[43(a+c)]=233(a+c)[43(a+c)]236
Done!


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