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|a−b|+2|b−c|+2|c−a|+5√ab+bc+ca
(choisiwon,Japan)
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=539142
Solution (thanks tian275461's hint)
We will prove that
P≥10√6
Without loss of generally,we can assume that a>b>c
by AM-GM inequality
2a−b+2b−c≥8a−c
which leads
P≥10a−c+5√ab+bc+ca≥4a−c10+√ab+bc+ca5
Thus,it's suffice to prove
a−c10+√ab+bc+ca5≤410√6=115√6
Or
(a−c)+2√(a+c)−(a+c)2+ac≤23√6
Now,Using Cauchy-Schwarz and AM-GM inequality,we have
(a−c)+2√(a+c)−(a+c)2+ac≤√2[(a−c)2+4(a+c)−4(a+c)2+4ac]=√2(a+c)[4−3(a+c)]=√2√3√3(a+c)[4−3(a+c)]≤23√6
Done!