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By this I mean books with math puzzles and problems similar to the ones you would find in mathematical olympiads.

  • 1
    as opposed to Martin-Gardner-type essays?2010-07-23
  • 0
    Community wiki?2010-07-23
  • 0
    Questions that are asking the user to create a list/collection of answers should be marked community wiki. I converted this question.2010-07-23
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    Olympiads are recreational to you? Dang.2010-07-30
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    everyone may play what what (s)he likes :-)2010-08-04
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    in my opinion, olympiad problems are far from recreational ones.2013-12-28

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If you're after Olympiad-level books, get The IMO Compendium which is a collection of problems from the International Math Olympiad, 1959-2004. You can find similar books with national Olympiad problems by going to Amazon and searching for "mathematical Olympiad".

Two books that offer collections of techniques useful for olympiad-level contests are Paul Zeitz's The Art and Craft of Problem Solving and Arthur Engel's Problem Solving Strategies.

There are lots of other books with similar titles and descriptions. Just follow Google Books's suggestions.

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Art of Problem Solving: http://www.artofproblemsolving.com/Store/index.php

Anything by Martin Gardner.

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    The Art of Problem Solving is very good for someone preparing for mathematics contests such as the AMC-12, but in my opinion too elementary for Olympiad-level training.2010-07-23
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THE PENGUIN DICTIONARY OF CURIOUS AND INTERESTING NUMBERS, by David Wells.

Sample entry: $199: 199 + 210n$ for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ provides the smallest $8, 9$ and $10$ primes in arithmetical progression.

Obviously you can reformulate any of the information given in the book as a problem to pose to someone else.