x,y,z>0 olsun $(x+y)(x+z)(y+z)-8xyz \geq0$ olduğunu gösteriniz

Question

http://www.matkafasi.com/5101/cemberinin-yaricapini-cemberinin-yaricapini-oldugununu

sorusu ile ilgili

Answer 1

$$x^{2}+y^{2}\geq 2xy\Longrightarrow x^{2}z+y^{2}z\geq 2xyz$$
$$y^{2}+z^{2}\geq 2yz\Longrightarrow y^{2}x+z^{2}x\geq 2xyz$$
$$z^{2}+x^{2}\geq 2zx\Longrightarrow z^{2}y+x^{2}y\geq 2xyz$$
O halde
$$x^{2}y+x^{2}z+xy^{2}+xz^{2}+y^{2}z+yz^{2}\geq 6xyz$$
Her iki tarafa $2xyz$ eklenirse%
$$\left( x+y\right) \left( y+z\right) \left( z+x\right) \geq 8xyz$$