-4
$\begingroup$

I did not win the International Mathematics Olympiad, so I was very curious to know the standard of questions asked in the graduate and undergraduate courses of maths and computer science in institutions like MIT, Stanford, Princeton, Oxford, etc. For example, suppose someone is taking a class in topology. Will the institution give problems related to the theory but the difficulty standard of the problems equivalent to that of an IMO or Putnam problem?

$\endgroup$
10
  • 12
    $\begingroup$ I'd like to reiterate what Steven Gubkin has already said in his answer here: Your questions on this site indicate a rather unhealthy obsession with the international maths olympiad. Studying mathematics at a university is quite different from solving problems in math competitions: the focus in a university math programme is on theory building, not so much on solving tricky but elementary questions. $\endgroup$ Dec 8, 2023 at 19:41
  • 5
    $\begingroup$ You cannot become a chess GM just by memorizing Magnus Carlsen's games. $\endgroup$ Dec 8, 2023 at 21:22
  • $\begingroup$ @JochenGlueck I always ask questions regarding imo standard problems because imo is focused on very few branches in mathematics but there are numerous other branches. So to get a good grip on those subjects we have to solve very tough problems ( the levels should be compared to Putnam ,imo and miklos) so that we can develop our mathematical skills. Please try to understand , I am looking for sources which provide very difficult problems regarding all the domains of maths. That is why I such questions in this forum. Thank you. $\endgroup$
    – Sillyasker
    Dec 9, 2023 at 18:23
  • 3
    $\begingroup$ I don't think solving IMO difficulty problems helps you understand any branch of math better! It helps you get better at solving hard problems, which is a separate skill than learning a particular branch of math. IMO is NOT about knowledge, but rather about problem-solving ability, which is a different thing. $\endgroup$ Dec 9, 2023 at 19:32
  • 2
    $\begingroup$ Here's a paper by Nick Trefethen on a certain kind of problem he used to give his graduate students: people.maths.ox.ac.uk/trefethen/publication/PDF/2011_137.pdf -- It's interesting, and since you mention Oxford, potentially relevant. $\endgroup$
    – user1815
    Dec 9, 2023 at 23:58

1 Answer 1

8
$\begingroup$

Will the institution give problems related to the theory but the difficulty standard of the problems equivalent to that of an imo or Putnam problem?

Probably not.

  • The whole point of a class (or textbook) is to get students to be able to solve most if not all of the problems that they are presented. Class problems are meant to be organized and scaffolded in a way that makes them fairly solvable if you paid attention to the underlying content and techniques covered in class or in the textbook.

  • The whole point of a competition problem (e.g. on IMO or Putnam) is for it to be very difficult to solve even if you know the underlying content. The goal is to "spread out" students' performance on the basis of their ability to think insightfully about these kinds of tricky problems.

As supporting evidence, just think about the typical grade distribution in a class vs the typical score distribution on one of these competitive exams.

  • In a typical class (in the USA, at least), the vast majority of students are getting 70%+ of the problems right on homework, exams, etc. Granted, if a class is covering really hardcore content, then the grades might be extremely curved (the most extreme case I've heard of was a something like 30% being passing and 50% being an A). But even still, this is no comparison for the typical score distribution on a competitive exam.

  • On the Putnam, for instance, the top scores are generally somewhere around 100 points out of 120 possible, and the median score is usually... wait for it... usually no higher than 2 points out of 120 possible. Typically the median is 1 point, and sometimes it's literally 0 points, meaning that if you scored any points, you did better than half of the people taking the test.

So, where can you find very very tough problems in graduate-level math? That's called research ;) After Putnam, there aren't really any more competitive exams. There are PhD qualifications, but those just serve to ensure that PhD candidates have a baseline level of expertise in their field. After undergrad, mathematicians don't compete on exams -- they compete on research impact. And as Raciquel rightly points out in the comments, the research phase is when the game begins to lend itself to more cooperative play.

$\endgroup$
7
  • $\begingroup$ could you please share with me any means or source where I can get very very tough problems regarding all the undergraduate maths topics like topology , linear algebra etc. I want to know the sources where I can find the problems related to every topics in graduate and undergraduate maths but the difficulty is that of Putnam , Miklos, Putnam etc. thank you. $\endgroup$
    – Sillyasker
    Dec 8, 2023 at 17:29
  • 1
    $\begingroup$ @SoumyadipDas to my knowledge, after Putnam, there aren't really any more competitive exams. There are PhD qualifications, but those just serve to ensure that PhD candidates have a baseline level of expertise in their field. Beyond that, people just focus on research, which I guess you could characterize as solving very very tough problems in graduate math. After undergrad, mathematicians don't compete on exams -- they compete on research impact. $\endgroup$ Dec 8, 2023 at 18:07
  • 3
    $\begingroup$ Upvoted. I think your third bullet point needs some qualification, though - or more specifically, a country tag. For instance, in math exams for first year undergraduate students in Germany the results are typically considerably worse than in your description, since the educational philosophy is quite different than, say, in the US. $\endgroup$ Dec 8, 2023 at 19:05
  • 2
    $\begingroup$ @JochenGlueck good point. Added a country tag to that 3rd bullet point. $\endgroup$ Dec 8, 2023 at 21:05
  • 3
    $\begingroup$ I would say that the point of a competition problem is to be very hard to solve even if you know the underlying content very well. In fact - the problems are meant to not depend too much on knowledge of content, so knowing the underlying content very well rather than just knowing the basics of the underlying content won't make it more likely for you to be able to solve the problems. $\endgroup$ Dec 9, 2023 at 23:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.