Question

Cho a,b,c > 0 . CMR :

1 < a/a+b + b/b+c + c/c+a < 2

Answer 1

Ta có : \(\dfrac{a}{a+b+c}< \dfrac{a}{a+b}< \dfrac{a+c}{a+b+c}\left(1\right)\)

\(\dfrac{b}{a+b+c}< \dfrac{b}{b+c}< \dfrac{b+a}{a+b+c}\left(2\right)\)

\(\dfrac{c}{a+b+c}< \dfrac{c}{c+a}< \dfrac{c+b}{a+b+c}\left(3\right)\)

Cộng từng vế của ( 1;2;3 ) , ta có :

\(\dfrac{a+b+c}{a+b+c}< \dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \dfrac{2\left(a+b+c\right)}{a+b+c}\)

\(\Leftrightarrow1< \dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< 2\)