Your mom has an undergrad degree from Northwestern? That will make you double (undergrad + grad) legacy, which is a pretty big hook. That will be a huge leg up. It sounds like your mom did residency/fellowship at UMich and WashU. The residency legacy doesn’t help as much. Your dad’s degree from Chicago Booth will certainly help, but not as much as undergrad legacy. It doesn’t matter that your interests are in business and your mom did the medical program. The main thing that matters is that your mom received her undergrad degree from Nwestern.
The effect of legacy hook has been revealed from the recent Harvard litigation. At Harvard, a legacy (from Harvard College) has a 7 x higher odds of acceptance as compared to non-legacy. This translates to a probability of acceptance of around 80%. For more details google the Arcidacono report and take a read. This advantage is not miniscule. The only factors which provide a greater benefit are: athletic recruit, being of a certain race, and being on the Dean’s interest list. Now each college handles things differently, so I don’t know how much weight Nwestern places on legacy. But the advantage is massive.
Sorry I don’t know much about math competitions and the timing of them.
Where ever you end up, best of luck! Please update the thread once you get your decisions.