Question

Fie x,y,z≥0.Sa se demonstreze ca :
$(x^2+y^2)(y^2+z^2)(x^2+z^2) \geq xyz(x+y)(x+z)(y+z)$

Answer 1

$\displaystyle Este~suficient~sa~demonstram~ca~x^2+y^2 \ge \sqrt{xy}(x+y). \\ \\ Avem~x^2+y^2 \ge \frac{(x+y)^2}{2}= \frac{x+y}{2} \cdot (x+y) \ge \sqrt{xy} (x+y). \\ \\ Si~asta~incheie~demonstratia,~caci \\ \\ \prod(x^2+y^2) \ge \prod \sqrt{xy}(x+y)=xyz(x+y)(y+z)(x+z).$