Question

CMR: 1<\(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}< 2\)

Answer 1

\(\left\{{}\begin{matrix}\dfrac{a}{a+b}>\dfrac{a}{a+b+c}\\\dfrac{b}{b+c}>\dfrac{b}{a+b+c}\\\dfrac{c}{a+c}>\dfrac{c}{a+b+c}\end{matrix}\right.\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}>\dfrac{a+b+c}{a+b+c}=1\)

(1)

\(\left\{{}\begin{matrix}\dfrac{a}{a+b}>\dfrac{a+c}{a+b+c}\\\dfrac{b}{b+c}>\dfrac{a+b}{a+b+c}\\\dfrac{c}{a+c}>\dfrac{b+c}{a+b+c}\end{matrix}\right.\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{a+c}>\dfrac{2\left(a+b+c\right)}{a+b+c}=2\)

(2)

Từ (1), (2) \(\Rightarrowđpcm\)