设a,b,c都是正数,求证:1/2a+1/2b+1/2c大于等于1/(b+c)+1/(a+c)+1/(a+b)急

问题描述:

设a,b,c都是正数,求证:1/2a+1/2b+1/2c大于等于1/(b+c)+1/(a+c)+1/(a+b)

用1/2a+1/2b+1/2c减1/(b+c)+1/(a+c)+1/(a+b)即可!
若正数,则1/2a+1/2b+1/2c>1/(b+c)+1/(a+c)+1/(a+b);
若负数,则1/2a+1/2b+1/2c<1/(b+c)+1/(a+c)+1/(a+b)。

利用基本不等式:1/x+1/y>=4/(x+y)
故有:1/4x+1/4y>=1/(x+y)
1/2a+1/2b+1/2c=1/4a+1/4b+1/4b+1/4c+1/4c+1/4a>=1/(a+b)+1/(b+c)+1/(c+a)