"If X is an integer, is Y an integer?
(1) The average (arithmetic mean) of X and Y is not an integer.
Insufficient.X and Y could be 1 and 2, for example.
(2) X + Y = 2 (X - Y)
Insufficient. If X is 1, Y is 1/3. If X is 3, Y is 1.
But, together, every combination of X and Y that satisfies condition 2 (examples):
Averages to an integer.
There are no (integer, integer) pairs that satisfy both conditions.
I say C.