Case-1:
AM of x,y and y-2 is x
i.e. [(x+y+y-2)/3] = x
=> [x+2y-2]=3x
=> 2y-2=2x
=> y-2=x
=> y = x+2
Thus if x is an integer, x+2 is an integer. i.e 'y' is an integer.
SUFFICIENT

Case-2:
(x+y)/2 is not an integer
Here x is an integer.
If (x+y)/2 is not an integer then y can be odd or y can be any non-integer value.
In either case (x+y)/2 will be a non-integer value.
Thus we do not get a precise value.
NOT-SUFFICIENT

Ans: A[/spoiler]