Đại úy
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,8\,1xyz\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,\,\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,(\,7\,8xyz\,+\,x^{\,3}\,+\,y^{\,3}\,+\,z^{\,3}\,)(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,+\,w\,)^{\,6}\,\geq\, 1\,024xyzw\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,+\,w^{\,2})\]
Thượng sĩ
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,8\,1xyz\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,\,\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,(\,7\,8xyz\,+\,x^{\,3}\,+\,y^{\,3}\,+\,z^{\,3}\,)(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,+\,w\,)^{\,6}\,\geq\, 1\,024xyzw\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,+\,w^{\,2})\]
$81xyz\left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2}} \right)\\ \le 27{\left( {xy + yz + z{\rm{x}}} \right)^2}\left( {{x^2} + {y^2} + {z^2}} \right)\\ \le {\left( {{x^2} + {y^2} + {z^2} + xy + yz + z{\rm{x}} + xy + yz + z{\rm{x}}} \right)^3}\\ = {\left( {x + y + z} \right)^6}$
Đại úy
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,8\,1xyz\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,\,\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,)^{\,5}\,\geq\,(\,7\,8xyz\,+\,x^{\,3}\,+\,y^{\,3}\,+\,z^{\,3}\,)(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,)\]
\[(\,x\,+\,y\,+\,z\,+\,w\,)^{\,6}\,\geq\, 1\,024xyzw\,(\,x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}\,+\,w^{\,2})\]
$$\left ( x+ y+ z \right )^{5}+ \left ( x+ y+ z \right )\left ( xy+ yz+ zx \right )^{2}\geqq 30\left ( xy+ yz+ zx \right )\left ( x^{2}y+ y^{2}z+ z^{2}x \right )$$
$$2\left ( x+ y+ z \right )^{5}+ \left ( x+ y+ z \right )^{3}\left ( xy+ yz+ zx \right )\geqq 63\left ( xy+ yz+ zx \right )\left ( x^{2}y+ y^{2}z+ z^{2}x \right )$$
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