The places near the South Pole are infinitely many. Some of them lie on the circle that is \( \left(1 + \frac{1}{2\pi} \right) \) km from the pole. Then you'll walk 1 km towards the pole, then circle it (1 km) and walk back 1 km in the same track away from the pole. Some other of them lie on the circle that is \( \left( 1 + \frac{1}{4\pi} \right) \) km, you do the same but you circle the pole twice. Thus the starting point could be the North Pole or any point that is \( \left( 1 + \frac{1}{2n\pi} \right) \) km from the South Pole, for any positive integer \( n \).