I was wondering if there was an area for writing contest, because I would like to try to enter some. I know that there is art contests so I assumed that there should be writing contest, but I haven't been able to them.

If you're not finding any, it's because there aren't any being run. :) Avatar contests are usually in the Avatar Chat section and Art contests are in the art forum. if there are any writing contests I imagine you could find them in the Community Contests forum in the Community Discussion area.