Question

You're at the center of the Earth and you go straight to 1km south and then you go to 1km to East now where are you NOW​

Answer 1

There is more than one answer in fact two (perhaps, strictly speaking there are an infinite number of starting points): I will give you the least obvious answer:

Your starting point is (1 + 1/(2pi)) miles from the South Pole. From this point, then, you walk 1 mile South. You are now 1/(2pi) miles from the South Pole. Now you walk 1 mile West, but since you will walk along a very high latitude, you will walk a circular path of circumference 2pi(1/(2pi)) = 1 mile. So your walk of 1 mile West is a complete circle. You then walk 1 mile North, which brings you back to your starting point, (1 + 1/(2pi)) miles from the South Pole.