Question

设0＜x,y,z＜1,求证x(1-y)+y(1-z)+z(1-x)＜1

Answer 1

证明:因为00,1-z>0,xyz>0

所以(1-x)(1-y)(1-z)>0,1>1-xyz

因为(1-x)(1-y)(1-z)=1-x-y-z+xy+yz+xz-xyx

所以1-x-y-z+xy+yz+xz-xyx>0

所以1-xyz>x+y+z-xy-yz-xz

所以1>x+y+z-xy-yz-xz

所以1>x(1-y)+y(1-z)+z(1-x)