一道高中竞赛题-业主生活–手机房天下问答

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躲在墙下等红杏丨Lv 2

设x,y,z为非负实数,证明 (x^2+y^2+z^2+yz+zx+xy)^2>=4(x+y+z)(xy^2+yz^2+zx^2)证明(1),记T=(x^2+y^2+z^2+yz+zx+xy)^2-4(x+y+z)(xy^2+yz^2+zx^2),则有4T=4(x^2+y^2+z^2+yz+zx+xy)^2-16(x+y+z)(xy^2+yz^2+zx^2)=(x^2-xz+3xy+y^2+3yz-3z^2)(2x-y-z)^2+(3x^2-3zx-7xy+3y^2+9yz+7z^2)(y-z)^2,证明(2),T=(x-y)(x-z)(x+y)^2+(y-z)(y-x)(y+z)^2+(z-x)(z-y)(z+x)^2>=0证明(3),用差分代换.

+1 2010-10-07 举报