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.
 
# (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) 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)

1. (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).
2. (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.
3. What is the maximum product you can obtain from a set of positive natural numbers that sum up to n.
4. (romanian national olympiad, 10th grade 2000) How many ways can you tile a 3xn rectangle with dominoes.
5. (romanian IOI selection 1999)

==code(c) |
  for i1 = 1,n
    for i2 = i1,n

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

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

6. How many ways can k rooks be placed on a nxn chessboard so that they don’t attack each other.
7. (romanian county olympiad, 11th grade 2000) How many permutations of length n have no fixed points. A fixed point for a permutation p is p(i) = i
8. ()(())
9. How many ways can you go from 0,0 to n, m.
non palindromes
10. black and white beads

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"
Every year during my highschool there was at least one problem in the national olympiad or in the ioi team selection tests from this book.
[2] Ioan Cuculescu "Olimpiadele Internationale de Matematica ale Elevilor"
[3] E. A. Morozova, I. S. Petracov, V. A. Skvortov "Olimpiade internationale de matematica"
[4] "Probleme de matematica traduse din revista sovietica KVANT"
[5] A. M. Iaglom, I. M. Iaglom "Probleme neelementare tratate elementar"

11 ramsey theory 3 people all know eachother 3 people don’t know
3a 3b 3c
permutation with max cycle p
permutation with largest p, ordonarea cu nr minim de swaps
f(f(x)) = f(x)