How to improve mathematics for Programming Contests?

I would divide the skills required for programming competitions as follows:

1. The ability to think abstractly.
2. The ability to employ well-known algorithms and algorithmic techniques.
3. Knowledge of previous competitions tasks.
4. Flexibility of point of view which allows to find easy implementations.
5. General problem solving techniques.

Skill (5) consists of ideas such as "backward reasoning" and "reversing a task" (these are not even specific to exact sciences). Skill (4) is an art which is gained mostly by experience and also by looking at work done by other contestants. Skill (3) is obviously a matter of experience. Skill (2) is attained by thorough preparation including both solving problems and reading algorithms books. Skill (1) is again an art, but this time it is an art which has much in common with other branches of mathematics.

Let me give you some examples from the latest International Olympiad in Informatics (IOI2011, http://www.ioi2011.or.th/tasks):

Task RICEHUB (relatively easy) requires an insight similar to task POST from IOI2000, which is an example of skill (3).

Task RACE (relatively hard) is mostly a matter of applying a known technique. The easiest way to solve this one is to try known techniques one after the other until you realize which one gives an efficient algorithm. I think it's very hard to solve this if you approach it any other way (I'm not giving details of the solution to avoid ruining it for you. There's a solution on the website). So this is an example of skill (2).

Task PARROT (relatively easy, but conceptually new in IOI) can be used as an example of skill (4). The "mathematical" solution is not too hard, but there are implementation issues. Again, in order not to spoil the question, I would just suggest considering task MAGIC SQUARES from IOI1996 and task TWOFIVE from IOI2001 which involve implementation details similar to the ones in PARROT (MAGIC SQUARES is easier than PARROT and TWOFIVE is harder in respect to those implementation issues).

As for skill (1), task CROCODILE from IOI2011 can serve as an example, but I'd rather give you a cleaner one which also serves as an example for skill (5):

Consider the well known egg-droping problem (see http://www.ocf.berkeley.edu/~wwu/riddles/hard.shtml).

To solve this problem one should first reverse it and ask "what's the highest building I can solve with up to $t$ throws?". To solve the reversed problem, one should employ backward reasoning (reversing and backward reasoning are examples of skill (5) ) and say: Let's try to sketch a solution strategy: The first throw would be from... well.. I don't know where from. So let's denote the floor of the first throw by "t_1" and continue from there. I will not continue describing the solution, but note that "let's denote the floor by a variable and continue" is an example of abstraction. Instead of a number of a formula, we just accept the fact that we don't know what this floor is yet, make up a variable to denote it, and continue.

(you may find tasks from IOI up to IOI2008 in http://ioinformatics.org/history.shtml).

From my experience with students participating in IOI, I would say that the ones who also participate in IMO (or otherwise very oriented toward mathematics and not just computer science) are:

• Always on top in regard to skill (1)
• Tend to be better than others at skill (5)
• Have a potential to become exceptionally good at skill (4) after some hard training

EDIT: As a final advice: Try solving many tasks. Code any task you solve to make sure you understand all the details. After solving (or trying long enough), always read an official solution or discuss the solution with someone else. In a more advanced stage of your training, read tutorials you can find on the web especially about data structures. You can find some of these on TopCoder's website.

As topics which are traditionally considered as mathematics and not computer science:

1. Mathematical games can be relevant to programming competitions.
2. In many computational geometry tasks it may be worthwhile to feel comfortable with convex sets as mathematicians think of them (maybe this is a bit advanced).
3. General mathematical experience and rigor can help you avoid wrong solutions which intuitively seem correct.

Overall, I'd say that the best way to become better in programming competitions is to train specifically for them while getting general mathematical education. There are not many "traditional math" topics which can help you directly with those competitions.

(probably the best and most relevant way to get general basic math education is to take "Linear Algebra" in university or to learn the same material elsewhere. Again, this is not directly relevant, but eventually your problem solving skills would benefit from the rigor you will learn at such a course).

As a slight counterpoint, I'll make the claim that if your goal is to become a better programmer, then the math skills you're looking for for contest problems probably aren't what you really want.

Don't get me wrong - I'm a huge fan of 'programmer math' (discrete mathematics, combinatorics, algorithmic analysis, etc) and believe that it's an immensely useful tool to have in your repertoire; and I absolutely adore 'recreational' mathematics and some of the branches of number theory that often wind up a part of programming contests. But those skills have virtually never been a part of my day-to-day work as a programmer (and I'm in one of the fields where they're most likely to come up), and in my opinion my knowledge of them is almost orthogonal to my programmer skills.

There's still ample use, certainly, for some of the basics - being able to simplify a problem first, being able to analyze the algorithms you use, all are handy skills; but the core tenets of programming (organization of code, abstraction, etc.) will IMHO carry you much farther. Programming contests are a lot of fun, and they're certainly encouraged, but they're not really a reflection of 'real-world' programming, in my experience, and I'd be a little leery of putting too much effort into skills that apply primarily to contest problems.