Question

Cho \(0\text{≤}x,y,z\text{≤}1\). Chứng minh \(x+y+z-xy-yz-zx\text{≤}1\)

Answer 1

Theo đề bài ta có:

\(\left\{\begin{matrix}x\ge xy\\y\ge yz\\z\ge xz\end{matrix}\right.\)\(\Rightarrow\left\{\begin{matrix}x-xy\ge0\\y-yz\ge0\\z-xz\ge0\end{matrix}\right.\)

\(\Rightarrow x+y+z-xy-yz-xz\ge0\)

Xét tích

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

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

\(0\le xyz\le1\) nên \(1-xyz\le1\)

Vậy \(x+y+z-xy-yz-xz\le1\)