Do$x\geq1 \Rightarrow x^{2}\geq x$
$\Rightarrow P\geq x(\frac{1}{(x+y)^{2}+x}+\frac{1}{z^{2}+x})\geq \frac{4x}{(x+y)^{2}+z^{2}+2x}$
gt$\Leftrightarrow (x+y)^{2}+z^{2}=3\left[ {(x+y)+z} \right]\leq 3 \sqrt{2\left[ {(x+y)^{2}+z^{2}} \right]}$
$\Rightarrow (x+y)^{2}+z^{2}\leq 18$
$\Rightarrow P\geq \frac{4x}{18+2x}=2-\frac{18}{x+9}\geq 2-\frac{18}{1+9}=\frac{1}{5}$
Dấu''='' xra$\Leftrightarrow x=1;y=2;z=3$