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I'm planning to write a math olympiad in a couple of months (4-5), and am just really trying to get the preparation in. I'm a fairly good math student (did ok in math, not an A+, but I got an A so my fundamentals are good I think). I was just wondering if someone could suggest some excellent books that are good for beginners and will open my mind to questions involving concepts such as:

-Telescoping sums (If $a_1a_2\cdots a_n=1$, then the sum $\sum_k a_k\prod_{j\le k} (1+a_j)^{-1}$ is bounded below by $1-2^{-n}$)

-Proving arithmetic progressions that meet some sort of criteria think these kinds of questions (Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite)

etc... and in general constructing proofs about sequences/series, that sort of thing.

That would be really awesome. I'm kind of scrambling all over the place to find the approximate syllabus for these Olympiad questions, so guidance in finding resources to learn would be great.

Thank you.

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Which olympiad is it?

All olympiads have become highly competitive. If you want to do well, then (1) you need to spend a lot of time practising, (2) you need to practise against a time limit. There are large numbers of olympiad problem books available now. And there are lots of problems available on the web.

But don't turn to the solutions too quickly. A good rule of thumb is that you need to struggle with one hard problem for at least half-an-hour every couple of days. It is no good getting into the habit of giving up and asking for help after 2 minutes.

You will be up against people who will be practising olympiad type problems for dozens of hours every week.

If you want more specific recommendations, I need more details about the particular olympiad and some feel for how good you are. For example, here is an old and easy IMO problem: A is the sum of the decimal digits of $4444^{4444}$ and B is the sum of the decimal digits of A. Find the sum of the decimal digits of B.

How long did it take you to do that?

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  • $\begingroup$ I am a 9th grade student, I finished Algebra 2, but in my spare time I also learned a lot of Precalculus and have it down fairly well (I know about arithmetic sequences, converging/diverging, that sort of thing...). I am practicing for the COMC (Canadian Open Math Challenge). It's not an Olympiad so to speak, but it's the qualifier I have to write in order to write the CMO (Canadian Math Olympiad). I've went through the Art of Problem Solving Probability & Intro to Algebra books and now I'm on Intermediate Algebra. $\endgroup$ – user164403 Sep 2 '14 at 18:03
  • $\begingroup$ And the answer to that old IMO problem is? :) But the obvious questions for you to practise on are old CMO problems. I have not looked at the last 10 years, but the early years were not too difficult. I have the questions for the first 36 CMOs. You should start practising. Try doing one question every two days. $\endgroup$ – almagest Sep 2 '14 at 19:31
  • $\begingroup$ I'm not at the IMO level yet, easy question or not easy... I know the question has something to do with number theory though, like modular arithmetic... That's all :X. I would really appreciate some book suggestions though, I feel sometimes as if I'm not at the level to understand the answers to some CMO problems. $\endgroup$ – user164403 Sep 2 '14 at 19:54
  • $\begingroup$ You cannot get far by reading. You need to practise (over a period of years) thousands of problems. There are not many general techniques. There are large numbers of small tricks etc. So there is no substitute for trying to do problems. Things seem easy when someone (or a book) explains them to you. But sitting in front of a problem is quite different. $\endgroup$ – almagest Sep 2 '14 at 20:02
  • $\begingroup$ Books like AoPS guide you through with problems though and greatly enhance your thinking skills by providing AMC and COMC-esque problems. AoPS books don't really target proofs and the kind of questions I posted here on this website. Despite looking at the excerpts and table of contents in Algebra & Intermediate Algebra, these topics aren't really discussed much. I'm thinking of taking a Coursera or EDX course on sequences/series or something like that... Otherwise I have no idea how to target these types of questions with proving. $\endgroup$ – user164403 Sep 2 '14 at 20:13

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