Up to now I always felt a little saddened when I see people drop out of the IMO or EGMO team selection. But actually, really I should be asking myself what I (as a coach) could do better to make sure the students know we value their effort, even if they ultimately don’t make the team.
Because we sure do an awful job of being supportive of the students, or, well, really doing anything at all. There’s no practice material, no encouragement, or actually no form of contact whatsoever. Just three unreasonably hard problems each month, followed by a score report about a week later, starting in December and dragging in to April.
One of a teacher’s important jobs is to encourage their students. And even though we get the best students in the USA, probably we shouldn’t skip that step entirely, especially given the level of competition we put the students through.
So, what should we do about it? Suggestions welcome.
Power Overwhelming 
An analogy that is reasonably accurate for sociological questions about this type of selection process is that of university admission to the very highest ranked programs. e.g., Harvard College, the Rhodes Scholarship, a Princeton theoretical physics PhD, or some other thing that is so selective that very few competitors can accurately predict their own success.
If “applicants” drop out of these due to an accurate perception that they lack the ability or the motivation to pass the selection, that just increases efficiency for all concerned. So to the extent that these are correct self-assessments that (in the case of US IMO) for whatever reason they will not make the team, or are not as interested in training/competing year-round as they had thought, then dropping out is not a bad thing and there is no reason to prevent it.
Inversely, if a lot of the dropout (what’s the rate?) is by the best people, defined as those who would have succeeded had they continued, that’s a sign the level of competition is set too high relative to the value of the prize, leading to adverse selection. Instead of the absolute top talents in the screening pool, you are selecting for people who are (in addition to high talent level) particularly competitive, persistent, (over)confident, have exceptional support, or other extraneous advantages. All those traits are good things but not what the selection is intended to accomplish.
I’ve obviously been out of the game for too long to know for sure what things are like today. But given the implausibility of the US IMO team matching or surpassing China on a regular basis given the difference in population size, incentives and education systems, I would guess that there’s a very good chance things have reached the point of hypercompetition and adverse selection. Another sign of that would be more top people dropping out of the elite layers of the AMC ecosystem earlier on in high school, to focus on other types of contests (research, programming, etc) or other math goals.
If that’s true then providing extra support could be considered as inadvertently enabling the problem, not mitigating or solving it.
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My comment got so long that I forgot to state the point of the analogy to university admissions, which was this:
The criteria used in those selections over time exert an evolutionary effect, presumably a negative one, on the population of applicants and even potential applicants. Students distort their lives and schedules, and the process not only attracts but creates ambitious resume builders — a skill set correlated with, but not at all the same as, the attributes the selections are meant to target and encourage.
With the IMO team selections, anything that artificially increases the amount of competition between (one nation’s) students, or is excessively tailored to the contents of the IMO rather than long-term mathematical skills, has potential to create this sort of adverse selection/evolution. I think it was good in a way for all the other countries to resign themselves to China winning almost every year at the IMO; if it’s because China trains too much or has an insurmountably large population base, then there is less incentive for other countries to fight a training arms race just to try to sometimes be number one.
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