I always ask my friends around me. There are so many people making music around one. It's also more fun to have music from someone you know. And they will feel honoured.
Then again, if you use it commercially, it would only be fair to pay them. But a video contest is not commercially, I would say.
If you don't know anyone, check out Soundcloud. There are many out there that would like to be part of an animation. I get even asked once in a while by musicians. They are looking for free animations....
incompetech is a great site, creative commons... search for KevinMacLeod, he is amazing!
His tracks require crediting him and are royalty free. Many You-Tube videos use his stuff.
There's also still a lot of free music at Jamendo including for commercial purposes. Unfortunately Jamendo itself seems to have gone more commercial and appears to be trying harder to sell it's own (admittedly reasonably priced) licenses, even for the music on it's site which still doesn't require a licence for commercial use. Check the Creative Commons terms.
There's a few links on the Creative Commons website: http://creativecommons.org/legalmusicforvideos/ but I can't recommend any of them specifically - I use production music libriaries and commerical music in my work.