Apparently treatments for PTSDs help reduce depression a great deal. You may also have an ADHD
Intelligence is highly valued. College will be different.
It's never as simple as having the best GPA. That is important but good colleges want someone who will "fit" in there also. Find some alumni to talk to. When I feel overwhelmed, it helps just to have a plan. To do something. You are brilliant - this is just like any other problem you've had to solve.
This problem has far more variables than a simple math problem. And, the results are not so easily quantified. But, it is the same kind of thing. The worst thing is you won't get into Princeton. It's not the end of the world but I know it seems like it and I don't mean to belittle your feelings. The people on those admissions boards are just people, too. They can be affected. Appeal to them - as yourself. Don't be whiny - just explain your goals and why you need this - be as objectively passionate as you can. In the end, if it isn't enough, at least you won't have to live with yourself knowing you didn't try every single thing you could.
I get what you're saying. What you need is a paradigm shift and, unfortunately, it will probably happen with or without your permission. I am truly sorry for that.
Even if the proofs already exist, you will still get a lot of credit for having figured it out on your own - at your age. For college, you really do have to sell yourself - and it is a lot more important than many job opportunities in life since this is it. Show them what you got - and that you are more motivated and willing to do whatever it takes. And, to do that well, you have to believe in yourself or at least be able to fake it well enough to fool a bunch of geniuses. :)
Most professors really do want students to succeed. Find someone local (or online, if necessary) to help you, if that's what you need. I promise you can find a math professor to help you out - hopefully someone in your chosen field. Just get out there and make it happen.
You just need someone qualified to check your work (okay, it's more complicated than that, but same idea) - at university, peer review is extremely important. It doesn't have to be the greatest person - but, you need someone.
Yes - exactly. Math doesn't have many females in it - so, they're going to automatically want to help you succeed. That includes Princeton's admission's board, too.
Fear is normal - but don't ever let it stop you from going after what you want.
Something is up. A life-changing moment. Even if it goes perfectly, it will still create anxiety. And... you're smart enough to know the truth of self-fulfilling prophecies. Right now, the point is doing everything you can to succeed. You can't control outside influences.
I'm familiar with that quote. :) It's true for that game but, out here, your success isn't determined by a psychopathic robot... I think. Just you. I think it will be hard for you to truly believe in yourself until you prove - to you - than you can succeed. Until then, do what every other adult does - fake it.
^.^ Point to you. But, the answer is still the same.
The hardest part is learning to compartmentalize. If you consider every possibility at once, you will definitely be overwhelmed. Look at the first problem, solve it and tuck it away; move onto the next one, and so on. There will be times when you have to back track - just handle them the same way. Accept the small successes as being just as valid as the big ones, too - because they are. Every large success is really just a cumulation of many small events. I don't know you very well but it's obvious you are smart and driven - and arrogant, which, in this case is a very good thing. Use that. Things become much more manageable when you believe in yourself. And, besides, it's not really arrogance if it's true. :)
Most people who are actually the smartest ones, tend to think they're about average. Why? Because many things that seem so easy to others are very difficult for them. But people with high IQ's do not think the way other people do. Being very good in math almost certainly puts you in that category. Having ADHD does make things harder, that is true. But, you are clearly driven enough to overcome it - you have been. Don't ever settle for something out of fear. No one ever knows what he/she is really capable of until tested. There are a lot of state tests, sure, but, the real test - is this right here. The fear.
You are mathematically intuitive. So, you can understand the concepts, but not necessarily the process others used to originally discover them... the process most people need in order to understand. University professors are a lot more likely to get and appreciate that fact than high school instructors. But, that doesn't help with the tests. Which brings us back finding someone to oversee your work in creating your resume. In college, you will find many more people who understand you and, with whom, you share commonalities. Right now, I'm heading to bed for the night, but I don't mind talking to you, if it helps. I'll be back tomorrow, most likely. Take care and try not to let yourself become too overwhelmed.