When you mail a letter, the post office, puts a dated "postmark" on it when they process it. So if your letter does not have a postmark that is dated Dec 31st or earlier, then it won't be accepted by the scholarship committee. Your best bet would be to get it to the nearest post office before noon tomorrow, just to make sure you get it out in time.
If the letter is stamped Dec 31st, then you are okay and the committee will accept it. It should have the dates post mark, which can only be put by the post office if you get the letter there in time. However, you will have to take it to a post office before tomorrow noon, to be sure.