How do I show others what I am capable of? I just turned sixteen this January, and I have started thinking about where I will go to pursue research in theoretical physics and mathematics. I live in India, and, sadly speaking, research here in India is awful, even though many claims of scientific advancements in India are made. First, I would like to share what I do at home.

I have mastered differential and integral calculus, and I have independently proved, and discovered some of my own proofs of theorems in Real Analysis(Proofs which I haven't seen in some other source, but they might exist). I have also independently discovered the well known techniques of solutions to certain kinds of differential equations, without having any knowledge of how to solve them. I continue to explore higher level mathematics, and try my best to prove each theorem I encounter myself. 

Now, the thing is, nobody at school, neither the teachers nor my friends, know anything about this. Even my parents are not fully aware about all this, but I try my best to share this with them. At school, the curriculum is such that the teachers are teaching to the test, and a heavy focus on contests and olympiads is laid, and to choose subjects of your own choice, you have to score high enough in subjects like Social Science and English, which don't even matter in research. Unnecessary debates and discussions are held, and I try my best to not share anything with these guys, to prevent getting into an hour long debate on what you should pursue.

In India, the major institutions are the Indian Institutes of Technology, Indian Institute of Science etc, where almost everybody focuses on getting into the IIT. To get an admission in the IIT, a standardised test, comprising of problems from Math, Physics, Chemistry and MAT has to be given. The admissions are solely decided by the score obtained in this test, which comprises of contest-like questions, which do not describe the research capability of the candidate. Even coaching centres have been set up all over the country, which train for this exam(called the JEE exam). The situation is really worse, and recently, I was demotivated by my teacher for not scoring enough in my social science exam. I have gotten used to this.

I would like to get admitted to institutions like MIT, Harvard, and I have read how I apply for them and what are the requirements for admission. The main thing is I have to distinguish myself from others, take part in extra-curricular activities etc, and maybe win math/science contests. But the thing is, I am not great at contests, which are timed, and I rarely would like to participate at school, where conditions are horrible. I have started taking leaves from school, because I think nothing productive is going on there. 

Now, I would like to ask if I apply for these institutes, how do I share what I am capable of? As a young mathematician, I think I am really doing well, and will do well in research too. But, other than my notebook, where I write all my thoughts and all my findings, I don't have anything that can show my abilities. If I just tell my abilities to them, like I did here, they would not believe in me, which I understand.

So, my question is, how do I show the institutes my abilities? Is it necessary for me to win contests at national/international levels, which is not that easy?
To apply for admission as an undergraduate at ... (MIT, Harvard, etc.) ... their web sites shows the steps you need.  There is an exam (SAT, Scholastic Aptitude Test) such applicants may take.  Probably there are opportunities to take it in your country as well, but maybe just in one or two cities, so you may have to travel there to do it.  (16-year-olds generally haven't published papers.)