Ta có: [tex]\frac{x^3-y^3}{x^2+xy+y^2}+\frac{y^3-z^3}{y^2+yz+z^2}+\frac{z^3-x^3}{x^2+xz+z^2}=0[/tex]
[tex]=>\frac{x^3}{x^2+xy+y^2}+\frac{y^3}{y^2+yz+z^2}+\frac{z^3}{z^2+xz+x^2}=\frac{y^3}{x^2+xy+y^2}+\frac{z^3}{y^2+yz+z^2}+\frac{x^3}{x^2+xz+z^2}[/tex]
=>[tex]=>P=\frac{1}{2}(\frac{x^3+y^3}{x^2+xy+y^2}+\frac{y^3+z^3}{y^2+yz+z^2}+\frac{z^3+x^3}{x^2+xz+z^2})[/tex]
Mặt khác ta lại có: [tex]\frac{x^2-xy+y^2}{x^2+xy+y^2}\geq \frac{1}{3}=>\frac{x^3+y^3}{x^2+yx+y^2}\geq \frac{x+y}{3}[/tex]
Tương tự ta cũng có:[tex]\frac{y^3+z^3}{y^2+yz+z^2}\geq \frac{y+z}{3}[/tex]
[tex]\frac{z^3+x^3}{x^2+xz+z^2}\geq \frac{z+x}{3}[/tex]
[tex]=>P\geq \frac{x+y+z}{3}=1[/tex]
Dấu "=" xảy ra khi x=y=z=1