Vryjdfkk

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!