Hello,

I answered this question as follows. Please correct me if I am wrong.

Needed: At least the formula of the circle.

S1: The circle intersects with parabola y = ax^2 + 4 where a>0 at only one Point.
This says Y intercept of the parabola is 4. It also says the Parabola opens upwards. Means the circle could be completely in second quadrant touching the parabola at one point or completely 1st quadrant touching the parabola at one point . Or could be just touching the Vertex of the parabola at y=4. The statement is insuff.

S2: The circle has center at origin
Clearly insufficient. The circle could be anything. Any number of individual integer points within circle possible.

S1+S2:
The circle is of the form x^2+y^2 = r^2 . Since the Y intercept of Parabola = (0,4) and the circle has to touch at only one Point - it has to pass through (0,4). Equation of circle - x^2+y^2 = 4^2. Number of possible integer x values = 9 (-4 to +4). Simmilarly there are 9 integer values of Y. Total possible integer values = 9*9 = 81. (Though this complete calculation is not needed for data sufficiency)

Final Answer : C[/quote]