Cho a,b,c,d >0 CMR: 1<$\frac{a}{a+b+c}$ + $\frac{b}{b+c+d}$ + $\frac{c}{c+d+a}$ + $\frac{d}{d+a+b}$ <2

@Magic_

Ta có :

$\dfrac{a}{a+b+c+d} < \dfrac{a}{a+b+c} < \dfrac{a+d}{a+b+c+d}$

$\dfrac{b}{b+c+d+a} < \dfrac{b}{b+c+d} < \dfrac{b+a}{b+c+d+a}$

$\dfrac{c}{c+d+a+b} < \dfrac{c}{c+d+a} < \dfrac{c+b}{c+d+a+b}$

$\dfrac{d}{d+a+b+c} < \dfrac{d}{d+a+b} < \dfrac{d+c}{d+a+b+c}$

Suy ra :

$\dfrac{a}{a+b+c+d} + \dfrac{b}{b+c+d+a} + \dfrac{c}{c+d+a+b} + \dfrac{d}{d+a+b+c} < \dfrac{a}{a+b+c}  + \dfrac{b}{b+c+d} + \dfrac{c}{c+d+a}  + \dfrac{d}{d+a+b} < \dfrac{a+d}{a+b+c+d} +\dfrac{b+a}{b+c+d+a} + \dfrac{c+b}{c+d+a+b} + \dfrac{d+c}{d+a+b+c}$ 

Suy ra :

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

Suy ra :

$1 < \dfrac{a}{a+b+c}  + \dfrac{b}{b+c+d} + \dfrac{c}{c+d+a}  + \dfrac{d}{d+a+b} < 2$ (đpcm)