Ta có$ay+az+bz+bx+cx+cy=(a+b+c)(x+y+z)-(ax+by+cz)$
BĐT tương đương
$ax+by+cz+2\sqrt{(ab+bc+ca)(xy+yz+zx)}\leq (a+b+c)(x+y+z)$
Aps dunjg bddt cauchy Schwarz
$VT\leq \sqrt{(a^2+b^2+c^2)(x^2+^2+z^2)}+\sqrt{2(ab+bc+ca).2(xy+yz+zx)}$
$\leq \sqrt{[a^2+b^2+c^2+2(ab+bc+ca)][x^2+y^2+z^2+2(xy+yz+zx)]}=(a+b+c)(x+y+z)$
ĐPCM.