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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?

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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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As a chess player I really like you comparison with chess ;) – Stefan4024 Feb 25 '14 at 0:41
@Stefan4024 yeah: there is definitely a lot of thinking and "gumption" strategies that transfer between the two disciplines :) – rschwieb Feb 25 '14 at 0:43
Yes, you are right. Maybe that's the reason why I find personal satisfaction in both chess and maths. – Stefan4024 Feb 25 '14 at 0:44

One of the best and biggest site for math contests is Art of Problem Solving. On its forum there are thousands of problem and they are nicely divided in category, so you can practise problems of specific area.

Another quite good site is Imo Math. All previous IMO math problems are there. Also if you spend some more time you can find other problem. Also sites like Brilliant are useful. They maybe won't provide a classic Olympiad problems, but there are lot of nice problems that develop you logic. Also there are lot of theory articles on site like these.

I personally prefer the method $(2)$, because there's no reason to spend $2-3$ hours on one problem. Even if you solve it you'll get nothing more than personal satisfaction. I would rather spend $15$ mins per problem and just try everything that first came to my mind. If it doesn't help me, see the start of the solution and return to the problem. If that doesn't work see the whole solution.

One of the main arguments for is that in my 10 years experience in math contests I have never ever met a problem that I have solved/met earlier. But, I have certainly have been familiar to the methods that will provide me the solution. Actually solving problems is used to get more familiar to the methods of solving Olypmiad questions and to build an intuition. Because, every contest problem is a combination of different methods and the more methods you know the bigger is the probability you know the "winning combination"

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I think the most important thing is to find right problem set to work on. Too easy or to hard will not help much. Your strategy 1) is not recommended, you will not improve your problem solving skill by just reading the solution without much tries and fails. So my suggestion is to use strategy 2), but start from easier problems, the ones that you can solve in one or two hours.

Even when you have to look at the solution, it's better to have someone (such as your friends who also love math) give you a hint first. Remember, Olympiad level math requires you to try every possibilities, and never give up is the key to success.

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My 2 cents (Tips):

  1. When you read a solution, try to figure out why the person decided to do that. Getting to a solution is never as simple as it is stated. The writer didn't just read the problem and "aha! this is how you do it!" it took some inspection and trial and error.

  2. Recognize patterns. No problem is unique -- you can always apply past knowledge to them. So next time you see, for example, a circle, you think power of a point. Next time you see an isosceles triangle, you think to draw perpendicular bisectors.

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