Argthtjtr

I am a graduate student in mathematics at a top school. Most of other students achieved gold medal in International mathematics olympiad for high school and/or other similar competitions. Most did their undergraduate in the very best school.

I came from a very modest educational background but worked really hard. I did not participate in IMO or other math competitions and if I did, I do not think I would have done well. I attended a not well-known small university in Australia.

Now that I am surrounded by all these gold medalists, I just feel that they are way beyond me and I am so dumb when I compare myself to them. How should I deal with this feeling?

I have thought about this too. I am also a math grad student. I went to a liberal arts college in the US. I took lots of non-math courses, didn't take any graduate courses, and didn't do any extra math activities like the Budapest semester. At some point I made a list of the math I learned as an undergrad and what I learned in grad school and it is a huge amount more in grad school. Another thing that helped was realizing that some of the grad students who, based on impressions, would do very well, never finished the program due to various reasons.

One thing that could help is to befriend grad students with whom you could let your guard down. There can be a lot of posturing, but much of it is quite obvious. Grad students who do that can still be helpful to talk to. It also helps to talk to faculty members whom you feel comfortable chatting with. They can really help you feel more at ease. I was at a workshop/school once and overheard a senior faculty member telling his math professor friend that by the end of a math conference, everyone feels like an idiot. So what if some grad students know more than you? At some point when you specialize you'll know a lot about quite a lot of things, and find joy in looking back at your accomplishments.

The fact that you are in the graduate program at all is a testament to the admissions committee believing that you are, at the least, just as good as those other students. It's normal to have the imposter-syndrome and you'd be surprised at how many of the supposed perfect students feel the same way, but are simply putting on a facade. I have never encountered any professor, graduate student, or colleague whom I had always regarded as natural geniuses mention that they had never experienced the imposter-syndrome. In fact, the ones who brag about how they believe they deserve to be in the program or department are usually the ones packing their bags the next year because they lost motivation to push through (from my experience, at least).

The key thing to keep in mind here is that, just like how you're idolizing one part of other students' accomplishments (e.g., gold medals), those same students may be idolizing some achievement of yours but are doing so without your knowledge for the very same reason of having imposter syndrome.

That being said, I personally do not think that having an imposter syndrome is necessarily a bad thing. This is what I always tell my undergraduate mentees who express similar feelings: The imposter syndrome is only as demotivating and negative as you let it be. Having those feelings can also be an indication that you're growing in a competitive environment and are able to see and acknowledge the areas where you lack compared to others. This just makes it easier for you to know what areas to improve on next. In other words, if you're smarter than everyone in your program, what room is there to improve? If you constantly think that you're doing an amazing job in life and don't need to pay attention to anyone else, are you actually improving at a competitive rate? Or are you already stagnant? Certainly this may be adaptive for self-protection in certain situations but I'd be surprised if anyone is able to keep up with the competitiveness of academia if they're only attentive of themselves.

I would say it's all about framing--feeling like an imposter certainly doesn't bring about positive emotions, but it can be a motivator for striving to produce positive outcomes. I am, however, a relatively cynical person who strives on negative emotions, so I would say that my personal take on the matter shouldn't override any kind of cognitive perspective on the matter that you find more useful...just as a late disclaimer.

First of all, let me assure you that a lot of people feel this way when they first join graduate studies. For some of them, this is the first time that they are truly challenged. Not all students go for IMO. I have a bachelor's and PhD in math and never attended one.
I felt small and insignificant in comparison to my peers, but then I started working on my problem. It took a lot of effort and reading and reasoning, and eventually I got a result! It takes all kinds of smarts to do math, and I think that once you get to know your peers you'll realize that you have a lot of worthwhile things to contribute!

Ask yourself this - why did you get into a top school? Those schools are pretty selective, and at least two people looked at your application carefully, in addition to your undergraduate lecturers who gave you sufficiently good grades to get in, your referees who wrote good things about you and so on. All of these people thought you're good enough.

Could all of these people be wrong about you?

Why do you think that you are in a better position to assess your quality than all of those experts who looked at your application and decided that you are good enough?

Good luck!

I have a slightly different take on this than the other answers:

It is a common experience for academics that, at school (as opposed to University), they were usually the brightest person in the room. At some point in their academic career, they will end up in a situation where they are no longer the brightest person in the room. This can be a profound shock to a person's self-image (you've always been "the clever one", "the swot").

It happened to me when I started at Cambridge University. It will have happened to all those people who went to "the very best schools". A big advantage of going to a place like that, is that essentially everyone is going through the same process at the same time.

The thing to remember is, that although you are now probably working in an environment where there are brighter people in the room than you, that doesn't mean you aren't entitled to be in the room.

You got your position. You performed well in university. You are going to be held to a higher standard but all you can do is remember what you have done and that you have shown you are ready for the next level.

I had the same thing going into undergrad. The main thing that helped me through was solving problems with others showing (to myself) that I could contribute. I gained confidence from that. As a postgrad I attended conferences with people from top ranked universities. Again I was worried I was not as good and only through talking to them about problems did I regain my confidence (thankfully they were nice). I have never met a grad student who wasn't worried about their ability to succeed, no matter what their past accomplishments.

You are doing maths. The main way you can get confidence imo is by solving what you have to learn. You don't mention how you are getting on with the work/any courses. How are you handling them? If you handle them (with plenty of work and study) then it does not matter how many gold medals someone has, you are matching up (indeed most graduate students I have met have been terrified about how they are getting on so it would be a great sign).

Maths olympiads are fun and it tends to show mathematical ability but it is not directly related to research. It is a different way of thinking and I wouldn't worry about them.

TL;DR: You can't control feelings. Shift the focus to yourself and what you can control which is your attention.

Treat feelings as signals, not drivers. You are better off just accepting their existence as they will not go away until you find and fix the source. In your situation, I'm guessing the the most essential source might be the fact that you dwell on comparing yourself to others.

Think of it this way: if you changed your environment and surrounded yourself with, say, data analysts (no offence to analysts, it's just a different world) - would you feel more empowered to pursue your goals? At least now you have the opportunity to learn from people you consider smarter than you what might prove more beneficial for you in the future.

The point is: comparing yourself to others provides no value. It's a silly counter-productive mechanism that distracts rather than motivates. Somewhere, there always be a person that's a few steps ahead of you.

Achievements vary from perspective. I couldn't care less about math medalists, to be honest. But this is a good thing actually, if you adjust your perspective to focus on making you achieve the goal, you won't care about other peoples' achievements, unless you are happy for them which I consider a healthy counter-mechanism. All you need to do is to keep the daily grind going and do what you need to do regardless of emotions. Motivation will probably appear sooner than you expect.

As a side-note, a good practice is to list all the achievements of the last year (not generally area-specific, it might be even something like going on a holiday) and remind them to yourself when you start overthinking the things you havent done. It has a better chance of putting you in a good mood than letting yourself down.

Disclaimer: This is a general advice that's based on my past experiences that are not academia specific, but are based on self-motivation towards my personal goals.