ok. so the summer olympics are in the southern hemisphere this year, and they’re occuring during the starting-to-get-cold months in the northern hemisphere.
if the winter olympics were to be held in the southern hemisphere, would they occur during the summer months in the northern hemisphere?
They’s probabily be held in spring, when it’s beginning to get warm up here and cooling down down there. The reason they didn’t hold this year’s olympics in November or December is because they wanted it to concide somewhat with summer up here.
If the Winter Olympics were in the southern hemisphere, there would need to be snow. Snow doesn’t hit the southern hemisphere until about May or June, if you’re lucky, but lasts until October maybe. Unless the whole thing is held in a mountain range, that is, in which case the snow-dates are a bit wider in scope.
No one has mentioned the obvious: we could have the winter Olympics in Antarctica any month of the year. 
Seriously, if you’re looking for a place in the S. Hemisphere with long winters, look for places with glaciers. The mountains along the Chile-Argentina border have glaciers. Puerto Natales, Chile (pop. 10,000), is about the only big town in the area. I know they have a lot of ski areas along the border, but I don’t know if they have any that far south.
South Island, N.Z., also has glaciers (but smaller ones) in the Southern Alps, about 100 miles west of Christchurch. Some people catch a ride on a helicopter or ski plane and ski down the glaciers around Mt. Cook. I believe the biggest ski areas in the country are on North Island, where the season ends in late November. One potential problem: Some of the ski areas on North Island are on the slopes of an active volcano.
There is also a smaller glacier (about 20 miles long) between Santiago, Chile, and Mendoza, Argentina.