Từ `x +y+z =1`
`=>` $\begin{cases} x + y = 1 - z \\ y +z = 1 - x \\ x +z = 1 - y \end{cases}$
Từ đó :
` (x+y)(y+z) +(y+z)(z+x) +(z+x)(x+y)`
`= (1-z)(1-x) + (1-x)(1-y) + (1-y)(1-z)`
`= xz -x -z +1 -xy -x-y +1 + yz - y -z +1`
`= xz + yz + xz -(x+x) - (y+y) - (z+z) + 3`
`= xy +yz + xz - 2(x + y+z)+3`
`= xy +yz +xz - 2 . 1 +3`
`= xy +yz +xz - 2 +3`
`= xy +yz +xz +1` ( ĐPCM)