557–
Unless Y can move away from X in a straight line, X can either cut the corner or turn inside the radius of the path Y is taking, thereby closing the gap and eventually catching Y.
Even if X and Y are both on the outer edge of the circle, X can still turn inside a smaller radius.
If there was an obstacle (such as an inner circle) within the circle or square, then Y could dodge X forever, since they would both circle that obstacle at the same speed.
Rough Rider–
Slice.
Shopping613–
It depends if the roach is the thing you went to get.
7 letters are in my name
If you subtract one 12 remains
What am I?
Twelves (or drytzen).