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best materials for IMO preparation?

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    I'd be grateful for some advice as to what books (or other materials) people would recommend as best for IMO preparation.

    Here's where I am at the moment. All comments are welcome!

    I am only looking at books which have solutions included. Therefore the otherwise excellent book by Monk, New Problems in Euclidean Geometry, is not included. Nor is Zeitz's Art and Craft of Problem Solving, although I don't think it's as good as Engel.

    best book: Engel, Problem-Solving Strategies

    and, other than that:

    best on number theory: Andreescu, 104 Number Theory Problems
    best on algebra: not sure yet (???)
    best on combinatorics: Andreescu, 102 Combinatorial Problems
    best on geometry: Bradley and Gardiner, Plane Euclidean Geometry, plus Andreescu's 103 Trigonometry Problems looks good although it doesn't cover the whole IMO geometry "syllabus"
    best on crossover areas: not sure, maybe relevant chapters in Andreescu's "Maths Olympiad Challenges"??

    UKMT books other than "PEG" seem not to match the above for usefulness.

    In coming up with the above list, I have looked at
    "USSR Olympiad Problem Book"
    "Primer for Maths Competitions"
    "Challenges in Geometry"
    "Maths Olympiad Treasures"
    all UKMT books,
    "Maths Olympiad Handbook"
    "Maths Olympiad in China"
    several other books with competition problems and solutions only, from the IMO, the US, etc.
    etc.

    One book I haven't laid hands on yet is The Art of Problem Solving, Volume 2: and Beyond. How do people rate this, relative to the alternatives?

    I've listed several titles by Andreescu. Would people agree with this, or are there other books which are much better, on the individual areas (especially number theory and combinatorics)?

    Thanks!

    owlie
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    AoPS Vol. 2 isn't really aimed at the same audience as the others. It's more of a textbook aimed at the AIME exam in the US, and isn't quite as advanced as the others mentioned.

    You've mentioned excellent sources, but there is only so far that books will take you with Olympiads, at the end of the day you have to do lots of problems, which is where http://mathlinks.ro helps a great deal. Its resources section is second to none.

    The only other book I'd mention is "Problems from the book" (another of T2's books, but meh, he's good)
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    Thanks very much for this. Nice to hear that someone else rates T2, and at last someone has told me what kind of book AOPS vol 2 is. Agreed about actual problems. I have got books of problems from China, USSR, US, UK, and the IMO. But because it's impossible to do everything, I'm limiting myself to IMO actual questions and shortlisted ones, referring to published solutions after solving them or after a few hours if I get stuck

    Any particular recommendations on algebra and geometry other than the above? T2 doesn't seem to be into geometry so much.

    owlie
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    (Original post by Owl_492)
    Thanks very much for this. Nice to hear that someone else rates T2, and at last someone has told me what kind of book AOPS vol 2 is. Agreed about actual problems. I have got books of problems from China, USSR, US, UK, and the IMO. But because it's impossible to do everything, I'm limiting myself to IMO actual questions and shortlisted ones, referring to published solutions after solving them or after a few hours if I get stuck

    Any particular recommendations on algebra and geometry other than the above? T2 doesn't seem to be into geometry so much.

    owlie
    I suspect the best texts for geometry are going to be created by the Easter bloc, but I never really cared for it so I don't really know anything about it.

    I never really "got" what Olympians meant by "algebra" either, so I can't really help you there.

    One thing I have found useful is "stalking" users on Mathlinks, reading just their posts to see how certain people thing. (darij and ZetaX being the natural ones to start with (Peter Scholze is also worth looking at)).

    That said, I was never very successful, so my advise should be taken with a pinch of salt
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    Thanks very much for this list!

    I havn't any information more than what you and SimonM have kindly given, but http://www.artofproblemsolving.com/ has been really useful to me in terms of finding information quickly and their large maths bookstore; perhaps you can fill in your gaps there.
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    In defence of AoPS V2, it isn't aimed solely for AIME or whatever (though many of the questions in it are similar to that competition), and is useful for your overall knowledge. Of course you can find the contents online, and it does teach some useful topics that other (especially UK books) neglect like logs and conics and Pell's equation. And some of the questions at the end are old IMO problems anyway.
    Try 'Geometry Revisited' and 'Plane Euclidean Geometry' if you are wondering about geometry.
    And go to the IMO compendium website; loads of resources on there.
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    Thanks for this. Only reason I didn't mention Coxeter and Greitzer's Geometry Revisited was that it isn't olympiad-focused. It's a fantastic and well-organised book.

    Would you say that Bradley and Gardiner's Plane Euclidean Geometry is sufficient, roughly speaking, for the "theory" required for the IMO? Can't quite work that one out.

    owlie
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    (Original post by Owl_492)
    Thanks for this. Only reason I didn't mention Coxeter and Greitzer's Geometry Revisited was that it isn't olympiad-focused. It's a fantastic and well-organised book.

    Would you say that Bradley and Gardiner's Plane Euclidean Geometry is sufficient, roughly speaking, for the "theory" required for the IMO? Can't quite work that one out.

    owlie
    I'm not sure whether the writers of Geometry Revisited intended to have such applications to Olympiad maths, but nevertheless it does. It actually contains more theory I'd say than PEG; but they work well together IMO-syllabus-wise.

    As for PEG, Chapters 3 and 4 are relevant to BMO1 and BMO2 (though both are of course useful for any competition), and nearly all the rest of the text is more suited to the IMO, however vectors and inversion for example have been used (as additional solutions) for BMO1 problems in recent years; and they appear later on in the book. However I wouldn't say it covered the whole IMO syllabus, but, alas, I'm not quite IMO standard yet so I'm uncertain as to how qualified I am to say that. But it does contain a lot.
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    ...which leaves algebra. With the constraint that according to my preference a problem-oriented textbook should contain solutions, would people say that the relevant chapters in Engel's Problem Solving Strategies and in Andreescu's Maths Olympiad Challenges are sufficient (plus loads of actual problems of course, but they're easy to find) for the IMO?

    I know there are many books and other sources from various countries which cover areas treated in IMO-y algebra (inequalities, functional equations, etc.) well in places. But I like to keep my number of textbooks down to a small number of outstanding ones, and they must have solutions. (So where algebra is concerned, this rules out the text by Naoki Sato for example!)

    owlie
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    Would you be able to recommend texts for the BMO syllabus?

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Updated: July 26, 2011
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