Question

Cho a, b, c > 0

CMR: a/a+b + b/b+c + c/c+a < 2

Answer 1

Theo đề bài ta có:

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

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

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

\(\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \dfrac{a+c}{a+b+c}+\dfrac{b+a}{b+c+a}+\dfrac{c+b}{b+c+a}\)

\(\Rightarrow\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}< \dfrac{a+c+b+a+c+b}{a+b+c}\)

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

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

\(\rightarrowđpcm\)