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Be creative and don't have the goal of solving the problem completely in mind. Have your enjoyment in mind. (4) Sleep on it, eat lunch, go for a walk, or crumple up the problem statement and dig it out from the trash in a month. Often the ability to solve a certain problem is based on what you have already learned.
This vision should be helpful to solve the most of usual problems. The base of a general method of hard problem solving is sketched in a book “Mathematical discovery: on understanding, learning and teaching” by George Polya. Should I just learn them "on the job"? I think this is a usual way. Why the origin of these tricks isn't explained?
I need this so that I can practice and fight with some most difficult problems. I believe that problems make us really stronger. It would be better if they are somewhat like Olympiad problems. I found a website known as project Euler- https://projecteuler.net. It consists of 590 difficult math questions. Any Help is appreciated. Thank You
I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could be related to things such as:
Finally, there is this list of $5$ interesting open problems by John. H. Conway (however, only $4$ of them are still open) All terminology used in the formulation of this problems is quite easy to understand. All objects mentioned here are usually mentioned on the undergraduate lessons too, and the ones that aren't are easy googlable.
12. Sometimes people work backwards. For example, here is an integral question I came up with while reading MSE: Prove that. ∫∞ −∞∫∞ 0 log(a2 + 1) (1 +x2)(1 +xa)(1 +a2)dxda =π2 log 2. Although, I think that many questions arise naturally at some point from research, or from thinking about other mathematical problems. – Eric Naslund.
This particular problem is quite simple for an Olympian. Since you can not solve it in 30 minutes, try doing simpler questions that are not Olympiad level. For example, the AMC and AIME series of questions are excellent. Simpler questions allow you to learn methods of solving problems in less time than doing Math Olympiad problems.
Try and understand it first and then write it down in your own words. Counterexamples, different approaches to the same problem or whatever that pops up in your head at that time, note them all down. That way you won't forget how you managed to come to an understanding the first time. (2) Don't let the work load pile up.
Aug 28, 2012 at 19:47. 2. Connecting the dots: 1) at night don't eat too much 2) just before goint to sleep try doing some physical exercise - perhaps short but intensive enough; this helps (me at least) to 3) sleep well and get up with more bodily/mental energy. – leonbloy. Mar 21, 2014 at 19:18.
$1.$ " $250$ Problems in Elementary Number Theory" - Waclaw Sierpinski. $2.$ "Problems from the book" - Titu Andreescu, Gabriel Dospinescu. $3.$ Any book provided by some university out there in England (I am sure there are plenty of handouts) $4.$ If you want to read about unsolved problems: "Unsolved problems in number theory" - Richard K Guy