Cho $ x, y, z> 0$. CMR:
$ 3\left ( x^{2}+ xy+ y^{2} \right )\left ( y^{2}+ yz+ z^{2} \right )\left ( z^{2}+ zx+ x^{2} \right )\geq \left ( x+ y+ z \right )^{2}\left ( xy+ yz+ zx \right )^{2}$
Cho $ x, y, z> 0$. CMR:
$ 3\left ( x^{2}+ xy+ y^{2} \right )\left ( y^{2}+ yz+ z^{2} \right )\left ( z^{2}+ zx+ x^{2} \right )\geq \left ( x+ y+ z \right )^{2}\left ( xy+ yz+ zx \right )^{2}$
Cho $ x, y, z> 0$. CMR:
$ 3\left ( x^{2}+ xy+ y^{2} \right )\left ( y^{2}+ yz+ z^{2} \right )\left ( z^{2}+ zx+ x^{2} \right )\geq \left ( x+ y+ z \right )^{2}\left ( xy+ yz+ zx \right )^{2}$
Ta sẽ chứng minh $x^2+xy+y^2 \geq \frac{3}{4}(x+y)^2$, hay
$$4(x^2+y^2+xy) \geq 3(x+y)^2$$
$$x^2+y^2 \geq 2xy$$
BĐT đúng. Do đó ta có $VT \geq 3 \prod (\frac{3}{4}(x+y)^2)=\frac{81}{64}(x+y)^2(y+z)^2(z+x)^2$
Cần chứng minh $\frac{9}{8}(x+y)(y+z)(z+x) \geq (x+y+z)(xy+yz+zx)$, hay
$$9(x+y)(y+z)(z+x) \geq 8[(x+y)(y+z)(z+x)+xyz]$$
$$(x+y)(y+z)(z+x) \geq 8xyz$$
BĐT đúng vì $(x+y)(y+z)(z+x) \geq 2\sqrt{xy}.2\sqrt{yz}.2\sqrt{zx}=8xyz$.
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