Question

xyz=1,求证：x/(1+y)+y/(1+z)+z/(1+x) >=3/2

Answer 1

楼上做错了!
楼上符号搞反了.
x/(1+y)+y/(1+z)+z/(1+x)
=（zx^2+yz^2+xy^2+2x+2y+2z+x^2+y^2+z^2)/(2+x+y+z+xy+xz+yz)
欲证x/(1+y)+y/(1+z)+z/(1+x) >=3/2
即（zx^2+yz^2+xy^2+2x+2y+2z+x^2+y^2+z^2)/(2+x+y+z+xy+xz+yz) >=3/2
即证2zx^2+2yz^2+2xy^2+4x+4y+4z+2x^2+2y^2+2z^2>=6+3x+3y+3z+3xy+3xz+3yz
即证2zx^2+2yz^2+2xy^2+x+y+z+2x^2+2y^2+2z^2>=6+3xy+3xz+3yz
x^2+y^2+z^2>=xy+yz+zx ……1(由(x-y)^2+(y-z)^2+(z-x)^2>=0得)
x^2+y^2+z^2>=3根号3次（x^2y^2z^2)=3…………2
zx^2+z>=2根号（z^2x^2)=2zx…………3
yz^2+y>=2yz…………4
xy^2+y>=2xy…………5
zx^2+yz^2+xy^2>=3根号3次（zx^2*yz^2*xy^2)=3…………6
把1,2,3,4,5,6相加就是
2zx^2+2yz^2+2xy^2+x+y+z+2x^2+2y^2+2z^2>=6+3xy+3xz+3yz
故原命题成立.
证完了,累死了,花了我好长时间.