I feel your frustration- both my kids really dislike competitions and avoided them throughout elementary and most of middle school. (Refused to do the spelling bees or the geo bee even when only a small group qualified, that kind of thing.) I think we also had issues with perfectionism, but also with sensitivity to the attention that comes with winning or showing your talents, some of which was related to bullying issues.
I don't know how math Olympiad is organized (we have math team here, but both kids balked and have refused to participate) but we did find success with the science Olympiad team- both have competed for several years, happily.
I think there are a few reasons this one was successful for us when other competitions were not- the SO is really team focused, and success is dependent in a large part on how well the team works together. Competitions are designed intentionally to require two or three students to work together, and their "practicing" takes this into account. So it is really collaborative- there is still pressure to succeed, but it is more about not letting your teammates down, not as much about individual pressure, if that makes sense. Plus, they loved being a part of a team, all the jokes, team spirit, etc, the community jail worked for my guys.
The other factor may just be the group dynamics- the SO team here is more collegial and they love to socialize, have fun etc- they are competitive, but learning and the peer group is the draw for most of them. We got the sense the math team was more about individual achievement, which for our kids was a big turn-off. There are a lot of parents who seemed over-invested in the results, and the school publicizes it, that kind of thing. Again, this atmosphere will differ everywhere, so you may just need to feel things out or talk with people involved to get a feel for the group, if you think that is a factor in your DDs behavior.
