Like all disciplines, the most important thing in competition math is practice. With very few exceptions, the people who perform at a level above their peers are those who have done more problems than everyone else. Competitive math is highly intuition based, and the simplest and most straightforward way to build intuition is to solve more problems.
Hence, problem dispensers are indispensable (pun intended):
Alcumus - Alcumus is a problem dispenser created by AoPS targeted towards middle school aged students looking to prepare for MATHCOUNTS and AMC competitions. It provides problems based on subject and level (1 being the easiest, 25 being the hardest).
MATHCOUNTS Trainer - MATHCOUNTS Trainer is a problem dispenser created by AoPS designed specifically to train students for the MATHCOUNTS competitions.
AMCTrivial - A personal favorite! This program is a problem dispenser made by searching the AoPS Wiki for problems. For more info on the difficulty scale, please refer to the AoPS Difficulty Ratings page.
Summer Programs:
For students who would like to better prepare for MATHCOUNTS state-level competitions, we highly recommend FrostMath (http://www.frostmath.com). Coach Frost runs online classes on Wednesday nights (6:15-8:45 pm) and in person classes on Saturday mornings in Lexington (9-11:30 am).
Students are also encouraged to prepare for and participate in IMLEM, MOEMS, AMC 8 and AMC 10.