[Math] Some Questions regarding preparing for Math Olympiads (searched but didn’t get answers)

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Many questions have been asked on this site regarding preparation for olympiads like the Putnam. I've read those questions and accordingly decided to start with Engel's "Problem Solving" but I have a few queries regarding how to practice the problems.

  1. Is Engel's book right for me? I think I have the knowledge of the material and that I just need lots of problems with clever solutions.
  2. How much time should I spend per problem? I am very bad at Putnam problems – I almost never get the solution – so is it better to: (1) spend just a little time on the problem, trying a basic approaches, and then looking at the solution, so that I can maximize my exposure to problem solving techniques in minimum time, or (2) spend a lot of time per problem and try as many possible approaches?
    I feel that (2) is very discouraging beccause even after spending time I usually don't get the answer, and that I end up spending lots of time on just 1 problem.

So far, studying has been a frustrating experience for me. I lose energy and feel discouraged when I start solving a problem because I almost always fail to do so. I lose my concentration and give up eventually. After looking at the solution, however, it seems so obvious. Is this just a phase that I'll get past, or am I studying wrong?

Best Answer

Here are some things to keep in mind:

  1. Not everybody solves every problem, nor solves every problem quickly.
  2. It is likely that you'll learn the most from trying to solve problems. You can probably learn helpful strategies about solving problems from "coaching" texts, but you are likely to learn (and retain) a lot more from practical experience.
  3. Don't "try to hard." Speed is not everything in life. As a rule of thumb, chess players get good at fast games by first getting good at slow games. I think mathematics is much the same.
  4. Try to "relax": the type of problems you are solving will usually be most efficiently solved by some insight. Ask yourself "does this look like anything I've seen before?" A lot of students struggle with problems when they are overthinking them.
  5. Another thing I transfer over from chess is to not get discouraged by slow progress. Problem solving is a skill that takes time to develop, and not everybody develops at the same speed.
  6. Another thing from chess: hindsight is $20/20$, so don't feel bad when solutions seem obvious. Hamilton struggled for a decade to multiply triples of real numbers to form a division algebra, but the disproof of this is an accessible exercise of undergraduate ring theory now.
  7. If you do eventually wind up reading a solution, try to reason to yourself how you could have figured that out. This is usually possible, and helps to build those problem solving muscles.
  8. Exposure to many problems is good. (This is actually more advice from chess.) The more you see, the more you learn and hopefully can transfer to other problems. This is one reason not to linger on problems too long. But it's not justification to give up on problems after thinking for only 15 minutes :)
  9. If you can find a teacher or friend that's skillful enough not to "spoil" problems for you, you might find it refreshing to exercise solving problems together.
  10. If you ever make a mistake (don't worry, you will) don't just brush it off and try to forget it. Think about what happened for a while and try to understand what went wrong, and you will probably be wiser the next time around. (coughchesscough)

From your rough description, it sounds like you are just feeling the initial discouragement that lots of people feel when first starting against a challenge. My advice would be to pick a fixed duration to struggle with a problem. Feel free to break the rules and extend it if you're having a good time. If I were you, I might spend an hour or two or more on a problem before looking for a hint. It depends on the problem. I would definitely avoid looking for a full solution for as long as possible. When practicing for qualification exams, I definitely spent an hour on many problems. One qual itself took $8$ hours to finish, and we were only submitting $8$ solutions. We all took the full time :)

Actually, think about moving on to other problems before looking up the solution to the last one. Sometimes you will find yourself solving the problem later, maybe the next day. There is no rule that you have to solve and understand them sequentially. I know for sure that in every math test I took, I jumped around doing different problems, and often felt like I was finishing the test more efficiently that way. Problems that were initially puzzling usually resolved themselves by the time I got around to them.

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