Short lists are often used in math camps to cover some subject by going through a bunch of problems. I've thought of doing the same for programming contests. My first list is related to combinatorics.
Feel free to discuss the solutions in the comment section.

# How many different strings of length 9 are there which contain 3 'a's, 3 'b's and 3 'c's.

# (olimpiada online 2001) Given n points on a circle. Join all possible n(n-1)/2 cords. What’s the maximum number of triangles one can see. Example: With n = 4 there are 8 triangles (4 with 3 of the 4 circle points and 4 with 2 circle points and 1 the intersection of the diagonal).

# (agora scholarships [4]) What is the maximum number of regions the plane can be split into by n lines. (Same problem for n circles, n planes, n spheres) Example: For n = 4 we can get 7 regions.
# [1] What is the maximum product for set natural numbers that sum up to n.
# (romanian national olympiad, 10th grade, 2000 [1]) How many ways can you tile a 3xn rectangle with dominoes.

# (agora scholarships) What is the maximum number of regions the plane can be split into by n lines. (Same problem for n circles, n planes, n spheres) Example: For n = 4 we can get 7 regions.
# What is the maximum product for set natural numbers that sum up to n.
# (romanian national olympiad, 10th grade 2000) How many ways can you tile a 3xn rectangle with dominoes.

# (romanian IOI selection, 1999)
  for i1 = 1,n
      for i2 = i1,n

               for ik = ik - 1,n
                   print ‘*’
How many stars will be printed for a given n and k.

# [1] How many ways can k rooks be placed on a nxn chessboard so that they don’t attack each other.
# (romanian county olympiad, 11th grade 2000 [1]) How many permutations of length n have no fixed points. A fixed point in a permutation p is p(i) = i
# (county level olympiad, 1994 [1]) How many correct bracket sequences of length 2n are there.

# How many ways can k rooks be placed on a nxn chessboard so that they don’t attack each other.
# (romanian county olympiad, 11th grade 2000) How many permutations of length n have no fixed points. A fixed point in a permutation p is p(i) = i
# How many correct bracket sequences of length 2n are there.

# How many ways can you go from 0,0 to n, m.

# Given a permutation of numbers 1 to n, what's the minimum number of swaps one can use to get to the identity permutation.
# Given a permutation p what's the minimum k so that p^k(i) = i for all i in {1..n}.
# (romanian ioi selection 98, topcoder 2004) Find a permutation p of n elements which maximizes k so that p^k(i) = i for all i in {1..n}.

# (acm.sgu.ru) black and white beads

# ladder 2 steps, 1 step

ladder 2 steps, 1 step

# ramsey theory 3 people all know eachother 3 people don’t know

# 3a 3b 3c
# Given a permutation p what's the minimum k so that p^k(i) = i for all i in {1,..n}.
# Given a permutation of numbers 1 to n, what's the minimum number of swaps one can use to get to the identity permutation.

Some math books useful for programming competition entusiasts:
[1] Ioan Tomescu "Probleme de combinatorica si teoria grafurilor"