Go to http://gc.kls2.com/
and try see for yourself.
EZE-PER is almost right over the pole, but is quite long -- 7900 mi.
A shorter route is PER-PUQ (Punta Arenas, Chile, the southernmost city in the world) -- 6600 mi.
EZE-SHA is over 12,000 miles - it's not much longer over the north pole!