Question

Cho a,b,c>0.CMR

a/b+c+b/c+a+c/b+a lớn hơn hoặc bằng 3/2

Answer 1

Đặt A=\(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)

\(A+3=\dfrac{a}{b+c}+1+\dfrac{b}{c+a}+1+\dfrac{c}{a+b}+1\)

\(A+3=\dfrac{a+b+c}{b+c}+\dfrac{a+b+c}{c+a}+\dfrac{a+b+c}{a+b}\)

\(A+3=\left(a+b+c\right)\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)\)

CM:\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{9}{a+b+c}\)(tự cm)

Áp dụng:\(\Rightarrow A+3\ge\left(a+b+c\right)\left(\dfrac{9}{a+b+b+c+c+a}\right)=\dfrac{9}{2}\)

\(\Rightarrow A\ge\dfrac{3}{2}\left(đpcm\right)\)