Hello, I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would be the books from The art of problem solving, Engels book and Paul Zeits book. Books on certain topics, say geometry is also appreciated!

Polya's "How to Solve It" is a good one. When prepping for the Putnam, I used "Problem Solving Through Problems" 


"Number Theory: structures, examples and problems", by Titu Andreescu and Dorin Andrica, contains many problems taken from the IMOs. 


Gelca and Andreescu have at least one book on the subject. See also the links to Math Circles and our own (University of South Alabama) list of suggested books. 


I enjoyed The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics by Shklarsky, Chentzov and Yaglom. 


Have you looked at the problems section in the Mathematical Reflections? It's a free online journal edited by Titu Andreescu. They publish six times a year and their problems tend to reflect current olympiad trends. Also, The Art of Problem Solving and Mathlinks.ro are message boards where olympiad contestants publish solutions from almost every contest in the world. Art of Problem Solving also publishes special textbooks geared towards olympiad students such as this one on Precalculus. 


See also Solving mathematical problems: a personal perspective by Terence Tao. 


For a slightly annotated list of some books on problemsolving, see http://math.mit.edu/~rstan/refs.pdf. 


I've read and enjoyed Putnam and Beyond. It covers more topics from algebra and analysis than I think are typically included in these books. Problems are selected from all sorts of competitions throughout the world including the IMO, various national selection tests for the IMO, and of course the Putnam itself. 


Titu Andreescu and Gabriel Dospinescu's Problems from the Book is new but quite nice. There are lots of beautiful examples in it proving a great deal of nontrivial results by what are essentially elementary methods. 


There is a book called 50 National Mathematical Olympiads in Slovenia published in English in 2006 by the Society of Mathematicians, Physicists and Astronomers of Slovenia at the occasion of the 47th IMO that took place in Slovenia. It contains all problems and solutions. 


Knuth's Concrete Mathematics is not only a fun place to learn great combinatorics; it also contains lots of amazing problems. 


High school level: Hungarian Problem Book I, II, III, IV University level: Contests in Higher Mathematics: Miklos Schweitzer Competitions, 19621991 


There are a lot of books my the american mathematical society. But I think the best book is Mathematical Olympiad in China by Bin Xiong. Any advanced geometry book will also help you because geometry is an area which really improves with practice. 


I prepared my Mathematical Olympiad with "The Mathematical Olympiad Handbook" by A. Gardiner. 


These are some of the books / links which I would recommend:



My favorite olympiad books were "Winning Solutions" by Edward Lozansky and Cecil Rousseau "Mathematical Miniatures" by Svetoslav Savchev and Titu Andreescu "Geometry Unbound" by Kiran Kedlaya [online notes] "Geometry Revisited" H. S. M. Coxeter and Samuel L. Greitzer Notes by PoShen Loh http://www.math.cmu.edu/~ploh/olympiad.shtml 


Hardy and Williams have "The Red Book of Mathematical Problems" and "The Green Book of Mathematical Problems", with some good analysis, series, conbinatorics, and group theory. de Souza and Silva have the more advanced "Berkeley Problems in Mathematics". And finally, another one I leaned on in studying for the Putnam was Dixon's "Problems in Group Theory". Although I do really like Polya and other's books on problem solving, I always found that I got more from them the more I actually worked through real problems. Others have mentioned actual problem books from the Olympiads and Putnam, which are of course great resources here as well. 


From a review for Polya's book on Amazon, the books to be read in sequence:
Full text of the review below:



I like "Geometry in Pictures" by Arseniy Akopyan and "Geometry of conics" by Arseniy Akopyan and Alexey Zaslavsky. http://www.mccme.ru/~akopyan/papers.html 

