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 22/11/2021 Bởi Kinsley 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) Bình luận
@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)