How does Antarctica stay frozen?
It doesn't stay frozen.  Ice evaporates (or sublimates is the correct term) under direct sunlight, but that's at a molecular level, it doesn't melt, it goes from solid to gas under sunlight and in the cold, some of this newly formed water vapor goes back being to ice.

In an absolutely dry climate, well below freezing, ice would slowly sublimate and disappear.  In a warm dry climate, ice would melt (much faster), then it would evaporate.   The sun does sublimate ice off the antarctic ice sheet every day, but quite a bit less than is added by snow and condensation from moisture due to temperature change and brought over by wind.   The antarctic ice sheet is also, always moving.   Mostly Antarctica loses ice by pushing it into the ocean - over long periods of time and it gains ice by (see above) snow and condensation.   The ice has been on Antarctica for over 30 million years, but as far as I know, the oldest ice core we've found there is about 1.5 million years old.    Everything moves.