Dropping weight on a fixed or single-speed bike is a great idea because it will let you be a *little* faster in a race that starts at the bottom of a mountain and finishes at the top. Some ways to drop the weight is to eliminate the extra cog and lockring, lose the rear brake, get rid of all reflectors, and change the seat post, stem and handlebar to those made of carbon fiber. Then you can change the stock pedals to eggbeaters by Crank Brothers or to the Speedplay Frog pedals. Dump the heavier stock bottom bracket and buy a top-of-the-line BB by either Campagnolo or Shimano. You'll save a good amount of weight with new wheelsets, preferably ones made with carbon fibers. Oh, and get the SLR 125 saddle.

This modification to your $500 bike should cost only south of $2,000, but you should be able to do well on that race up the mountain. Good luck. Please post pictures!