First things first: participating in this contest does
And if you want a programmer on you team, you can always form a team with one at our introductory meeting on January 24th!
Professor Spear used this Android framework in his ENGR 98 class. It was designed by Prof. Spear to introduce people to Java who never programmed before. In a couple of weeks, these students all went from zero programming experience to having fully functional Android games! Prof. Spear provides an excellent getting started guide on the Github wiki. No Android phone? No problem! Prof. Spear will be renting out Android phones for use during the competition at no cost.
Update: Here are the slides from the LOL workshop: Download
GameSalad is a fantastic drag-and-drop tool for developing iOS games. Their website provides an excellent set of tutorials telling you how to get starated. GameSalad also exports to Android and Windows 8, but unfortunately those options require a relatively expensive Pro account. Thankfully, iOS and HTML 5 exporting is free!
Construct 2 is another neat drag-and-drop tool that requires no programming. There's a great free version that allows you to export Windows 8 games, but you'll have to pay a significant sum to export iOS or Android games. Scirra's site has a ton of great video tutorials to help you on your way.