Question

[Kimian Hajan Ruventaren]

Cho a,b,c>0. CMR

\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{c+a}\ge\dfrac{a+b+c}{2}\)

Answer 1

[Nguyễn Trọng Chiến]

Áp dụng bđt Cô-si:

\(\dfrac{a^2}{b+c}+\dfrac{b+c}{4}\ge2\sqrt{\dfrac{a^2}{b+c}\cdot\dfrac{\left(b+c\right)}{4}}=2\sqrt{\dfrac{a^2}{4}}=a\)

Chứng minh tương tự :

\(\dfrac{b^2}{c+a}+\dfrac{c+a}{4}\ge b;\dfrac{c^2}{a+b}+\dfrac{a+b}{4}\ge c\)

\(\Rightarrow\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}+\dfrac{1}{4}\left(2a+2b+2c\right)\ge a+b+c\Rightarrow\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge a+b+c-\dfrac{1}{2}\left(a+b+c\right)=\dfrac{a+b+c}{2}\) Dấu= xảy ra \(\Leftrightarrow a=b=c\)