Problem:
For all positive numbers x,y,z, the product (x+y+z)−1(x−1+y−1+z−1)(xy+yz+zx)−1[(xy)−1+(yz)−1+(zx)−1] equals
Answer Choices:
A. x−2y−2z−2
B. x−2+y−2+z−2
C. (x+y+z)−2
D. x1yz1
E. xy+yz+zx1
Solution:
The given expression equals
==x+y+z1(x1+y1+z1)(xy+yz+zx1)(xy1+yz1+zx1)x+y+z1(xyzxy+yz+zx)(xy+yz+zx1)(xyzx+y+z)(xyz)21=x−2y−2z−2.