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)

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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)

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)
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)

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)
1/2a+1/2b+1/2c
=1/4a+1/4b+1/4a+1/4c+1/4b+1/4c
=(a+b)/4ab+(a+c)/4ac+(b+c)/4bc
又因为(a+b)/4ab-1/(a+b)=(a-b)^2/(4ab(a+b))>=0
故(a+b)/4ab>=1/(a+b)
同样的有(a+c)/4ac>=1/(a+c)
(b+c)/4bc>=1/(b+c)
所以(a+b)/4ab+(a+c)/4ac+(b+c)/4bc>=1/(b+c)+1/(a+c)+1/(a+b)
得到1/2a+1/2b+1/2c>=1/(b+c)+1/(a+c)+1/(a+b)