Question

Cho a,b,c là những số thực dương . CMR

\(a^2+b^2+c^2\le2\left(\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\right)\)

Answer 1

\(\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}=\frac{a^4}{a^2+ab}+\frac{b^4}{b^2+bc}+\frac{c^4}{c^2+ac}\)

\(\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^2+b^2+c^2+ab+bc+ac}\ge\frac{\left(a^2+b^2+c^2\right)^2}{2\left(a^2+b^2+c^2\right)}=\frac{a^2+b^2+c^2}{2}\)

\(\Rightarrow\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\ge\frac{a^2+b^2+c^2}{2}\Leftrightarrow a^2+b^2+c^2\le2\left(\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\right)\) (đpcm)

\("="\Leftrightarrow a=b=c\)