証 明 abc=8 1/a+b+2<都分母> + 1/b+c+2 +1/a+c+2小魚等魚2分ㄓㄧ高中 回應 s87009812 Re: 難題 2006-07-16 18:34 AM>=GM>=HM. abc=8, a+b+c >=3( abc )^1/3 = 6 a+b+2 >= 8-c 1/a+b+2 + 1/b+c+2 +1/a+c+2 小於等於 1/(8-a) + 1/(8-b) + 1/(8-c) HM = 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)] 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)] 小於等於 6 1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )小於等於1/2 所以1/a+b+2 + 1/b+c+2 +1/a+c+2小於等於1/2
s87009812 Re: 難題 2006-07-16 18:34 AM>=GM>=HM. abc=8, a+b+c >=3( abc )^1/3 = 6 a+b+2 >= 8-c 1/a+b+2 + 1/b+c+2 +1/a+c+2 小於等於 1/(8-a) + 1/(8-b) + 1/(8-c) HM = 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)] 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)] 小於等於 6 1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )小於等於1/2 所以1/a+b+2 + 1/b+c+2 +1/a+c+2小於等於1/2
回應
s87009812
Re: 難題
2006-07-16 18:34abc=8, a+b+c >=3( abc )^1/3 = 6
a+b+2 >= 8-c
1/a+b+2 + 1/b+c+2 +1/a+c+2 小於等於 1/(8-a) + 1/(8-b) + 1/(8-c)
HM = 3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )]
3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)]
3/[1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )] 小於等於 (1/3)[24-(a+b+c)] 小於等於 6
1/( 8-a ) + 1/( 8-b ) + 1/( 8-c )小於等於1/2
所以1/a+b+2 + 1/b+c+2 +1/a+c+2小於等於1/2