Question

[Nguyễn Minh Phương]

 Cho \(0\le x,y,z\le1\). CM: x+y+z-xy-yz-zx\(\le1\)

Answer 1

[Hoàng Phúc]

Có \(x\ge xy;y\ge yz;z\ge xz\)

=>\(x-xy\ge0;y-yz\ge0;z-xz\ge0\)

=>\(x+y+z-xy-yz-xz\ge0\left(1\right)\)

Xét \(\left(1-x\right)\left(1-y\right)\left(1-z\right)=-\left(x+y+z-xy-yz-xz+xyz-1\right)\ge0\)

=>\(x+y+z-xy-yz-xz\le1-xyz\)

Mà \(0\le xyz\le1=>1-xyz\le1=>x+y+z-xy-yz-xz\le1\left(2\right)\)

Từ (1),(2) có đpcm