I'll go against the other answers and say it helps, 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.

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.