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Scared about attending the Ross Math Program this summer

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The program is designed so that anyone with any background can approach the problem sets. My experience there was that other participants were very friendly and competitiveness was not really an issue.

Also from personal experience chess and other games were barely played. I'm not sure where you heard this from but doing non-pset related activities like playing games is actively discouraged.
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Hey, I was an attendee of the Ross program in 2021. The camp is incredibly friendly and there are people from all stages in their math learning. Some students really were gobsmacking  and awe inspiring but none were intimidating. It’s a very friendly community that rewards effort over brilliance. If you have never done competition math sort of problems before, it will be a different way of thinking but the counsellors are there to make the transition as smooth as possible.

Don’t be worried about completely solving the problem sets, some problems aren’t even meant to be solved completely but you are meant to make as much progress as you can on them. If you were chosen for the program, you are certainly ready for the rigor.
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Hi, it isn't important to be able to successfully solve problems, what is important is going from a situation of being totally lost (as we all are) to being slightly less totally lost. There isn't a unique way of doing that, which is why there are nearly disjoint areas of math. Human camraderie can go a long ways towards a feeling "this makes sense" starting from where you look at a solution and even if you understand why one line implies the next, it just seems meaningless and there'd be no reason to remember it. If you're talking to someone and don't understand something, don't be afraid to keep insisting that it doesn't make sense to you, and trying to describe what doesn't make sense. That itself is the main skill. Just to give one example from my past, if someone said, "The distance from 0 to 1 is 1" I'd be worrying, one what? One inch? One foot? one mile? Another example, someone would say

  1 2 3

(            )

  3 2 1

denotes "replacing" 1 by 3.  But does that mean I have a row of  3 coins and where a coin had the number "1" written on it, I erase that and write "3"?  Or does it mean I pick up the coin that has the "1" written on it and place it in the 'location' where the coin with "3" written on it had been? And in that case what is a "location" and why isn't the "location" encoded anywhere in the notation?
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You don't have to like chess just because everyone else around does. Some of these camps/programs have culture like that. But once you come to uni, you'll meet all sorts of mathematics students. Some play chess in their free time. Others like to drink. Some are really into sport. Some stereotypes are true about lots of them, but definitely not about all and whatever is your thing, you'll find similar minded people around you.
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As a past participant, it's all about putting in the time/effort. Ross is the sort of camp for people who (think they) will enjoy doing challenging math every waking minute for the whole duration of it. Presumably you knew that when applying and demonstrated that in your application.

Now, obviously there are limits to that. Some of your time will be spent eating (obviously) and possibly exercising. Some of it probably will be spent with other games/distractions like chess as you say (given it's a summer camp for youth and all), but that's not the point of it. The point is to learn math/mathematical reasoning, and hopefully you enjoy that process enough to stay engaged for most/all of the program's duration.
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Sounds like you are exceptional anyway. I say what is the fun in not trying?

Thank the people that support what you do by doing your best with the opportunity you are given.

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