The Sprint Round consists of 30 problems to be completed in 40 minutes. Calculators are NOT allowed, nor is the help of your teammates--you're totally on your own here. The format of this round is similar to that of conventional tests that you take in class.
Your score on the Sprint Round is the most important factor in determining your individual ranking. So it wouldn't hurt to put a little extra emphasis on it during your preparations.
The thing about the Sprint Round is that you get THIRTY problems dumped on you all at once. Compare this to the Target Round (2 problems at a time), the Team Round (10), and the Countdown Round (1). Basically, this means that you're responsible for pacing yourself on the Sprint Round. You want to go slowly enough to avoid careless mistakes, yet quickly enough so that you don't run out of time. How in the world does one find this perfect pace?
Practice is key. (Sigh, it always seems to come down to that little word "practice.") Make sure you practice at least one (and preferably more!) full-length, timed Sprint Round from a previous year. This will help give you a sense of your pacing. 30 problems in 40 minutes means an average of 80 seconds per problem, but you should go more quickly through the easier beginning problems. Wear a watch so that you can track yourself, both during practice and during the actual competition.
Don't be afraid to skip problems! The last thing you want is to lose points from a bunch of easy problems, just because you spent too much time on one hard problem. You can always come back to a skipped problem later, and by then your subconscious mind might have generated some new insights.
Don't be afraid to guess, either. There's no penalty for wrong answers. A few minutes before time is up, fill in all your empty blanks with guesses. After all, ANY answer is more likely to be right than NO answer. I believe that the proctors are supposed to warn you three minutes before time is up, or something like that. Sometimes they forget to do so, though--again, bring your own watch.
When you skip or guess on a problem, put a little mark next to it, like a star. Also mark answers that you're unsure of and want to check. That way, if you have extra time at the end, you'll be able to come back to those problems easily.
Most of all, don't quit! The Sprint Round is the longest round, and that can make it tiring and frustrating--sometimes, I think it should be called the Marathon Round. Toward the end of the 40 minutes, it can be tempting to finish early and take a nap. Don't do that! They call you Mathletes for a reason. Think how embarrassing it would be if a sports star said, "Oh, this game is too tough, and I'm tired. I think we should just give in a quarter early."