Kaggle is sponsoring a March Madness competition to predict the NCAA Tournament. (Although right now there are no prizes offered.) Some semi-useful data is provided at the contest, including that game outcome data for the past five seasons and tournaments. The data has been anonymized so it's usefulness is limited outside of the contest.
There are two stages to the contest. Stage One, which is ongoing right now, is to predict the results of the last five tournaments. Stage Two will be predicting this year's tournament when it starts up in March. (Obviously Stage One is meaningless -- it's trivial to un-anonymize the data and make perfect predictions.)
I don't see any clear indication of how the contests are being scored. Predictions are confidence levels, and the FAQ says that all games in the tournament will be equally weighted, but it isn't clear how submissions will be scored. I've submitted a topic in the forum asking that question.