Welcome to Android Central! You don't have to be logged into your AC account there, because you enter the contest either by using your Facebook account or by giving your email address.
What browser are you using, and are you on the computer or your mobile device? Make sure you're on your computer, and make sure Javascript is turned on. There should be a window that pops up when you click on the Use Your Email button. If Javascript is turned off, that might not work.