整体替代法证明.设a,b,c>0,证明：1/(2a)+1/(2b)+1/(2c)>=1/(b+c)+1/(a+c)+1/(a+b)

Question

整体替代法证明.
设a,b,c>0,证明：1/(2a)+1/(2b)+1/(2c)>=1/(b+c)+1/(a+c)+1/(a+b)

Answer 1

(a-b)²=a²-2ab+b²≥0
a²+b²≥2ab
(a+b)²=a²+2ab+b²≥4ab=======>(a+b)/(ab)≥4/(a+b)(此处用到a,b>0)
1/a+1/b≥4/(a+b)
同理1/a+1/c≥4/(a+c)
1/c+1/b≥4/(c+b)
三式相加为2/a+2/b+2/c≥4/(b+c)+4/(a+c)+4/(a+b)
两边都除以4即得