What topics should one study to become good at solving mathematical questions in Programming Competitions?

If you are taking part in the contests of all the mentioned websites and are able to solve all the other non-math-algorithm-intensive problems, then I would suggest that you study http://books.google.co.in/books/about/Mathematical_Circles.html?id=8bXkoKSbN9YC&redir_esc=y . It is an excellent text covering most of the mathematics that is covered in these contests. However, this book does not covers probability and other unconventional topics. The foundation that this book gives is so strong that you feel like tackling more and more difficult problems. However, this book has some truly difficult problems ( usually at the end of the respective chapters ) but the greatest strength of this book is the availability of the solutions to most of the problems at the end of the book. After studying this book ( or together with it ), you can tackle the problems given on http://codeforces.com/ by the topics that you cover in the above topic. If you work diligently on this book, I think that you should be able to attempt many problems by the time you finish half of the book and solve many problems by the time you complete the entire book. However, remember that this is a starter and some problems in the contests are pretty hard and use complicated mathematics. So, you'll have to study advanced texts on graph theory, probability and computational geometry as well. Don't worry, going through the above book will give a very strong foundation to tackle all these problems. All the best.

Note : This is not my own answer. This is combination of some other 2 - 3 answers on quora. If you feel there are still more to add, please suggest edits. I am happy to update :) Number Theory: Basics: LCM and GCD (Extended) Euclidean algorithm Modular arithmetic (addition, subtraction, multiplication) Modular multiplicative inverse ("division") - Existence and computation Sieve of Eratosthenes and Sieve of Atkins ( for Fast Prime list generation ) Fast prime factorization (with/without pre-process) Euler’s totient function. Exponentiation by squaring (e.g., computation of "A power n" for a square matrix A) Intermediate: Solving systems of linear modular congruences - Chinese Remainder Theorem Solving linear recurrences by fast exponentiation of matrices (e.g., Fibonacci Sequence) Euler's phi function and fast computations Miller–Rabin primality test Advanced: Primitive root modulo N Discrete logarithm Discrete square root Multiplicative function and Möbius inversion Formula Farey sequence and applications And also Logarithmic Exponentiation, Fermat’s theorem Maths: 1.    Linear Programming (Simplex) 2.    System of Linear Congruence 3.    FFT 4.     Number Theory concepts(listed at the end) Geometry :

I'm not quite all about competitive programming [as the matter of fact, I'm not at all], but you might check these out: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/ http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0CCoQFjAB&url=http%3A%2F%2Focw.mit.edu%2Fcourses%2Felectrical-engineering-and-computer-science%2F6-042j-mathematics-for-computer-science-fall-2010%2Freadings%2FMIT6_042JF10_notes.pdf&ei=kkUuVdDtF8GYsgGjs4CICA&usg=AFQjCNFjLJJ3KXhJgWfakTBMLZ2BU-W5Bw&sig2=XHN0rmiscnr0WumgGFGouA - written by some pretty clever guys working at Google [Eric Lehman], MIT/Akamai [F Thomson Leighton] and MIT [Albert Meyer]. 557 pages [didn't expect it to be easy, right?] Free for download - be advised: download starts automatically when you click the link. If you have any problem with that, please, do not use this link. Thanks for a2a.

As far as i know, computer programming need good ability in algorithm building and logical reasoning rather than certain critical Mathematics. Because most of the coding involves flow of certain process in efficient way estimating risks involved in building it. A coder should always be able to assess the parallel flows associated within a process. So a coder should have good grip in building a logic for achieving some process, hence rather certain mathematics topics, I would suggest in building a skill set with logical buiding, reasoning, writing algorithms. Of course it is a part and parcel of the coder to deal with certain numbers sometimes which doesnt really require any core mathematics knowledge. And again i want to specify this as well. As you progress your carreer as a developer, you might need some knowledge about handling some big numbers involving complex calculations.

Firstly there are better detailed answers to your question on Quora you could have searched as this is a generic question. To be honest this list will keep on growing as you start learning them. But for a start this will be sufficient. Matrix exponentiation Binomial coefficient Factorial - Factorial, Small factorials GCD - GCD2 Modular Arithmetic Number Theory Graphs Prime Numbers, Factorization and Euler Function Basics of combinatorics Miller–Rabin primality test Recurrence relations Inclusion–exclusion principle Coin toss problems Linearity of expectation value : Sieve of Eratosthenes and Sieve of Atkins ( for Fast Prime list generation ) Fast prime factorization Pigeon Hole Principle

I am not a pro programmer but I have participated in these competitions before.  According to my experience you must be strong in algebra and probability. Check out this link -- http://discuss.codechef.com/questions/37684/learn-competitive-programming

Combitronics is the most important one. You must know how to count.