x=y=z=2
You choose three POSITIVE integers: x, y and z.
The quotients 1. (x*y)/(x+y)
2. (x*z)/(x+z)
3. (y*z)/(y+z)
shall also be integers.
So, one has to prove the integers x, y and z have a factor in common which is bigger than 1.
The problem is that I absolutely don't know how to prove this.