Does the International Math Olympiad help research mathematics?

I think of Olympiad problems more as "parlour tricks". They're really difficult, and it's super-impressive if someone's good at them, but the skills are very different to the skills you need in research. As a big example of a difference: the Olympiad rewards quick accurate leaps of reasoning, because you're under such time pressure. Research rewards long-term grit and persistence through blind alleys and repeated failure.

Disclaimer: I participated in the International Math Olympiad, and have a PhD in operations research, which is essentially a type of applied math.

There is some overlap between math olympiads and research math. However, as others have noted, mathematics is a very broad field, which includes subfields such as: algebraic topology, theoretical computer science, combinatorics, control theory, optimization, statistics/machine learning. The amount of overlap with math olympiads depends very much on what subfield of "research math" you are referring to. Math olympiads have more overlap with say combinatorics, and less with say control theory.

In training for math olympiads, I learned how to try special cases to get intuition about how a problem works, and how to simplify a problem step by step, and how to write a logical and complete proof. I also learned how to persevere, and to enjoy the challenge of tackling problems which are difficult, and also when to give up when sometimes I am just stuck. I think that these are skills that also are required in research math.

Therefore, I think that there is positive but imperfect correlation between performance at math olympiads and performance in research math. If you don't do very well at the IMO, you can still be a successful pure math professor; and if you get a perfect score at the IMO, that does not mean you are guaranteed to have a successful research math career.

Finally, the math olympiad is an artificial competition, in that the problems in the olympiad can all be solved in a fairly short time with a relatively small set of tricks. On the other hand, in the real world, research math is much more open ended, you need to find and define your own research problems, and oftentimes the problems cannot be solved!

I would use the analogy that olympiad math is like an RPG such as The Legend of Zelda: Breath of the Wild, whereas research math is like real life, extremely messy and open ended.

I'll go against the other answers and say experience with math Olympiads helps a student to become a better researcher, although in a limited way. Math Olympiads give you a larger "bag of tricks", with which you can solve faster the easy-medium problems that you may encounter during your research.

Moreover, you arrive at university with a larger math background and understanding, struggle less with the material and are probably more likely to retain what you see in the lectures. For instance, separating the nontrivial ideas from the tedious details is much easier when you already have a lot of experience in solving problems and writing proofs.

And in maths, everything you know can suddenly become useful in another field. It's useful to have already seen something. As a numerical analyst, I have on occasion used ideas from other fields in my research: combinatorics, algebra, inequalities...

That said, Olympiads tend to produce "problem solvers" rather than "theory builders", and some students burn out after doing maths for so many years (but it's a very small minority) or lose focus in the lectures at the undergraduate level because they find them not challenging.

Disclaimer1: I have been an IMO contestant twice, and now I am heavily involved in the organization of the Italian Math Olympiad.

Disclaimer2: all of this is anecdotal (but so are all the other answers I have read up to now). I don't know if there is any rigorous statistical investigation on that.