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. x1​yz1​
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.​