a,b,c为不相等的正数，且abc=1
求证：根号a+根号b+根号c<1/a + 1/b + 1/c
參考答案:证明:
1/a+1/b+1/c=(ab+bc+ac)/abc
=ab+bc+ac
=(1/2)[(ab+bc)+(ab+ac)+(ac+bc)]
≥(1/2)[2(ab*bc)^(1/2)+2(ab+ac)^(1/2)+2(ac+bc)^(1/2)]
=(abc*b)^(1/2)+(abc*a)^(1/2)+(abc*c)^(1/2)
=b^(1/2)+a^(1/2)+c^(1/2)
所以
根号a+根号b+根号c<1/a + 1/b + 1/c
得证.
