@Chris45Well after thinking a bit more about it I now see there...

@Chris45

Well after thinking a bit more about it I now see there is a whole series of answers generalising my Part 2 answer, which I am hoping is your Part 3 answer.

There are a series of latitudes on which to locate the tent that meet the conditions. They are a distance North of the Soutn Pole equal to:

10+ 5/nPi

Where n is any integer from 1 to infinity.

The method is as before:
Say for example n=2, then proceed from a latitude 10+5/2Pi North of the South pole, Walk South 10 Km, then West for 10Km - that will consist of two circles round the South Pole ending back where the Westward walk started. Finally Walk North 10Km back to your Tent.

If N=3 you perform 3 Westward circles of the globe etc.

Finally at the absurd limit of n=infinity you walk 10Km to the South Pole then walk on the spot turning anticlockwise an infinite number of times, then proceed north 10 Km back to your tent!

Is that what your friend had in mind!