Question

Cho: \(\left\{{}\begin{matrix}a,b,c>0\\a^2+b^2+c^2=abc\end{matrix}\right.\)

Tìm Max:

\(P=\dfrac{a}{a^2+bc}+\dfrac{b}{b^2+ac}+\dfrac{c}{c^2+ab}\)

Answer 1

\(P=\dfrac{a}{a^2+bc}+\dfrac{b}{b^2+ca}+\dfrac{c}{c^2+ab}\)

\(\le\dfrac{a}{2a\sqrt{bc}}+\dfrac{b}{2b\sqrt{ca}}+\dfrac{c}{2c\sqrt{ab}}\)

\(=\dfrac{a\sqrt{bc}}{2abc}+\dfrac{b\sqrt{ca}}{2abc}+\dfrac{c\sqrt{ab}}{2abc}\)

\(\le\dfrac{2a^2+b^2+c^2}{8abc}+\dfrac{2b^2+a^2+c^2}{8abc}+\dfrac{2c^2+b^2+a^2}{8abc}\)

\(=\dfrac{4\left(a^2+b^2+c^2\right)}{8abc}=\dfrac{1}{2}\)