Hello, what are the fields of work in computer science that are most related to competitive programming. I mean the fields that need similar problem solving skills, and in order to do it, one should solve similar problems? (by competitive programming I mean something like IOI, ICPC, Google Codejam, Facebook hackercup..etc)

Beginning to intermediate competitive programming skills really apply to any programming job, since they have to do with language fluency, problem-solving, and basic algorithms. As you get more advanced in competitive programming, the skills you need start to become more specialized -- specific math and algorithms that tend to come up in competitions. Most programming jobs don't need those kinds of specialized skills, but big software companies will generally have relevant positions -- e.g., http://en.wikipedia.org/wiki/Petr_Mitrichev, one of the world's top competitive programmers, works on search at Google.

Questions like this come up a lot on the Competitive Programming topic on Quora (http://www.quora.com/Competitive-Programming). For example:

http://www.quora.com/Do-you-find-skills-you-achieved-during-programming-contests-to-be-useful-outside-of-them

http://victorliew.quora.com/Is-Competitive-Programming-Experience-Necessary-To-Be-A-Great-Developer

http://www.quora.com/What-are-the-practical-benefits-of-competitive-programming