Q&A: Are Homework Problems at Top Colleges as Hard as Competition Problems?

Cross-posted from here.

Question

I am 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?

Answer

No.

• 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.