I know this isn't exactly what you were looking for, nor does it have anything to do with FIRST, but
Topcoder has a very well designed, free, weekly on-line programming competition. You are given 75 minutes to write three functions that return the requested values. You can compete in Java, C++ C# or VB and are scored based on time used coding each problem and the difficulty of the functions.
They also have component competitions where you can win money by designing an entire component. This is closer to what you are describing as you are given a week(I believe) to create the program and each competitor's program is given a score by a panel of judges based upon coding ability and functionality.
I hope you find this helpful.
P.S. If they ask for the member that referred you(I'm not sure if they still do), my user name is j_johnso
Josh