\(P=\dfrac{1}{2}\left(\dfrac{2\sqrt{bc}}{a+2\sqrt{bc}}+\dfrac{2\sqrt{ac}}{b+2\sqrt{ac}}+\dfrac{2\sqrt{ab}}{c+2\sqrt{ab}}\right)\)
\(P=\dfrac{1}{2}\left(1-\dfrac{a}{a+2\sqrt{bc}}+1-\dfrac{b}{b+2\sqrt{ca}}+1-\dfrac{c}{c+2\sqrt{ab}}\right)\)
\(P=\dfrac{3}{2}-\dfrac{1}{2}\left(\dfrac{a}{a+2\sqrt{bc}}+\dfrac{b}{b+2\sqrt{ca}}+\dfrac{c}{c+2\sqrt{ab}}\right)\)
\(P\le\dfrac{3}{2}-\dfrac{1}{2}.\dfrac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}{a+2\sqrt{bc}+b+2\sqrt{ca}+c+2\sqrt{ab}}=\dfrac{3}{2}-\dfrac{1}{2}.\dfrac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}=1\)
\(P_{max}=1\) khi \(a=b=c\)