The easy solution is the north pole. However, this is not the only solution!

Imaging you start somewhere near the equator. You walk 1 mile south then you turn west. If you were to walk west for a long time, you would eventually return to the same point you are currently standing. You could then walk north to your original starting location. Of course, you would walk much farther than 1 mile west, but this gives you an idea. As you walk west, you would trace out a circle of some radius R and circumference 2*pi*R. If this circumference is an integer divisor of 1 mile, you could finish your westward trek at the same point you started.

For an infinite number of solutions, find a circle around the south pole of radius 1/(2*pi*N), where N is some integer. Let point A be any point 1 mile north of this circle. Walk south for 1 mile. Now walk west, following that circle, N times. Now that you've walked 1 mile west, return north to point A.