How do you manage to study and have a balance in your life at the same time?
Yes, it matters that you are in grade 9. It is probably a good thing that you are able to do what you like to do (math and computing), but it seems too early to focus yourself quite so much on narrow interests. The problem that might arise is that you get burned out before you are 20 years old with no backup plan for your life and also poor health and no relationships with people. That would be sad.
It is good to work intensely at something, but not quite as good to do so to the exclusion of other things. There is a lot to learn, and you are recognizing that now, at least a bit. In fact, making time for those other things, such as exercise (aerobic especially) and relaxation (reading, talking, ...) can actually make the intense study that you do more effective. At least this is true for most people, though there are exception. But, for most, continuing to push hard against an intellectual problem, say in math, can just waste time and lead to a deeper block, than taking a break. It turns out that, again, for most people, the brain/mind will continue to work on putting connections together when you are engaged in other activities, even sleeping.
But, for someone as young as yourself, I'd suggest that you think about the following priorities. First, your health, both mental and physical. Next, your relationships, both family and friends. Next, getting a lot of experience of different kinds, studying lots of things with deep study of a few of them. You seem to have the deep study part mastered, but may be missing out on the wide experience that can make the deep study more meaningful and also give you options to change direction as you grow older.
It is good to have some direction in your life, but not, yet, a good idea to exclude all other potential paths that you might follow.
As for time management, you can do it formally, or informally. If you joint a sports team or other directed activity, then your schedule will be set for you. But you can also just decide that you will ride your bike for an hour every day - preferably with a few friends. Or you can just decide, even more informally, that when you get stuck on a math problem that you take a walk in the woods for half an hour, or read a chapter in a book. Something to provide a bit of variety.
At a later stage in your life you will be forced by circumstances to specialize in something. It is good if you put that off for a while so that you have the opportunity to make the decision about where to specialize with more background knowledge and experience than you now have.
Time management and reasonable expectations. You need to determine what, specifically, your goals are, and why. It sounds like you enjoy what you are learning, but you are letting learning take over other aspects of your life. This is bad almost always. Here are my suggestions:
You shouldn't be studying everything every day. This is for many reasons, but one of them is that you literally will learn better if you don't. Our brains need time to process and contextualize information. This is why sometimes leaving a problem for a while and coming back to it leads to a solution you didn't see before. Say you drop one thing per day, so now you're only doing 2 per day, not 3. That frees up an hour and 20 minutes per day, and it gives you time to think.
With the free time, again, prioritize. You will now have about an hour to do something else. Perhaps you choose to exercise three times per week, do some de-stressing (read, play a game) twice per week, and see friends twice per week. Don't try to do it all at once.
Finally, we get to reasonable expectations. Regardless of your age, excelling at something is not equivalent to doing it as much as possible. Practice doesn't make perfect - it makes permanent. That means a few things:
- If you are practicing the wrong thing, you get really good at making mistakes.
- If you spend all of your time learning things in isolated fields, you will become a walking book.
- It's not what you put in that matters, it's what you get out.
Point 3 bears more discussion. Just because you are implementing algorithms doesn't mean you are learning anything. If I play the piano but I stick with pieces I've played since I was 6, I don't get better at piano. Choose to focus on very challenging problems. For instance, if your algorithm can play flappy bird for 2 minutes without failing, what, specifically, is required to get it to play for 10% longer? Are you focusing on really solving the problems, or are you just using someone else's code and seeing what happens? If it's the latter, don't waste your hour and twenty minutes, it won't help you in your goal.
Top-tier scientists in machine learning (for example) understand both the algorithms they work on as well as the problems they are trying to solve. They are technical experts, but that doesn't mean that they are hermits working away forever without any access to the outside world. Having a passion is important, but don't become your passion - be a well-rounded person with exceptional expertise. This means that you will need to accept that your progress will be slower than it could theoretically be if you only dedicated yourself to your studies. I assure you that every "top-tier" researcher I've ever encountered has had something that isn't their central problem to solve.
I think that's one of the things you will need to set an expectation for. You seem to think that it is impossible to be a top-tier researcher if you don't exclusively focus on your studies. Try...not doing that, as I suggested above, and really, honestly evaluate your progress.
To add on to the other two excellent answers, I want to say that you perhaps are putting too much on your own plate.
number theory(for olympiad and not applied), geometry(again, for olympiad), algebra(olympiad again), grade 10 math, competitive programming and machine learning
This is way too much to learn at the same time, especially since you are still going to school. Even undergrad students don't do so much at the same time: they might take a few classes on different topics concurrently, sure, but they have the luxury of time. All or most of the classes they take will be on these topics that they are studying, which I bet isn't the case for you.
I would suggest picking one thing to do at once, and doing it well. You can't learn effectively within time constraints if you're doing five different topics at the same time. Your brain needs time to digest new information that is learnt, and you have to give it space to do so. If your brain has no space to breathe, you'll not learn effectively, not to mention that it will (needlessly) take away time from other things in life while not giving you anything good to make up for it.
In addition, don't just study a topic because you feel like you "should" (e.g. because that's what's tested in Olympiads), do it because you like it. Try out a bunch of different things and try to see what you like, instead of doing it for the sake of doing it. If you already have a strong passion for everything you're currently doing, great! Keep that up and also be on the lookout for other things on the horizon. Otherwise, find a passion in one particular item and stick to it for a while. (Note: I'm not suggesting that you abandon everything else, but you can't do everything at once. Focus on this one thing that seems interesting at the moment for a while at least--perhaps a few weeks. But don't get sucked in so much that you ignore everything else.)
Practically: don't feel "obliged" to study four hours a day, in the fields that look like you "have to" study. Just pick one thing you like, and do it as a hobby. Stick to it for a while, but it you ever get disinterested, don't be afraid to switch. You'll come out liking math better, and you'll be much better at it too (ironically). Be careful not to get absorbed, or it will suck away the rest of your life, as you observed!