Yes you should
This is your big break to win money.
Sure you might not be very pretty, but you can at least try. You got a recommendation and that means something!
I highly encourage you to fill in your application. If you don't, you'll regret it. And may haunt you for the rest of your life.
Besides..even if you don't win, at least you will have fun doing it.
It's not always about winning, it's about having fun. Try your hardest, but always remember, "Have fun."
So yes, I think you should sign the application, and join in the fun.