I have heard many say that being able to solve Olympiad problems is by no means a prerequisite to becoming a good mathematician, physicist, etc. However, would one benefit from practicing math competition problems if he is older, i.e. undergrad level and on. Would there be any benefit to the person by being able to solve the problems in unrelated fields or for solving unrelated problems as a researcher or on an unrelated exam?
I did some Olympiad problems as an undergrad when I found a book on it in the Math Library at Rutgers, I liked some of the problems too. In my opinion anything that challenges you, helps make you sharper, or teaches you something new is always worth while.
I personally wouldn't do too many olympiad problems if I was out of high school (which I'm not, so maybe I'm mistaken) unless I was preparing for putnam or doing it for pure recreation. It doesn't seem to be very relevent to undergrad/grad math. Of course they're a great way to develop your ability to solve problems in high school (or for putnam but then again there are plenty of putnam problems to do) but I doubt an undergrad will ever run out of math to do which is relevent to what they're studying.
I was planing to read some introductory math competition book that also contained problems. My intent is not to do many Olympiad problems, but rather i was hoping to gain better ability to solve problems.