Check out these problems from the 2006 AMC 8.
#17 – This is an excellent probability problem that deals with how the sum of odd numbers are created.
#20 – This counting problem looks at a single elimination tournament and the number of games won/lost by different competitors. It takes a good bit of logic to get to the solution:
#22 – This is also a good logic problem involving numbers:
#24 – Another logic problem involving multiplication: