can giai gap ho bai toan .

H

hiensau99

1/ Ta có:[TEX] \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}> \frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{b+ c+a}=1[/TEX]
Ta có: [TEX]a^2<a^2+cb \Rightarrow a^2+ab+ac<a^2+cb+ab+ac \Rightarrow a(a+b+c)< (a+b)(a+c) \Rightarrow \frac{a}{a+b}<\frac{a+c}{a+b+c}[/TEX]
Chứng minh tương tự ta có: [TEX]\frac{b}{b+c}<\frac{b+a}{a+b+c};\frac{c}{c+a}< \frac{c+b}{a+b+c} \Rightarrow \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}<\frac{a+c}{a+b+c}+ \frac{b+a}{a+b+c}+\frac{c+b}{a+b+c} =2[/TEX]
Vậy: [TEX] 1<\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}<2[/TEX]=> ĐPCM

2. đã có ở: http://diendan.hocmai.vn/showthread.php?t=121336
 
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