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, Arthur Engel's book and Paul Zeit's book. Books on certain topics, say geometry, are also appreciated!

1$\begingroup$ But why does MO need such questions? If MO needed them then MO should split into subfora: the MO proper, soft, education, history, philosophy, ...; and of course Meta. Yes, I have proposed a split on Meta a time ago (more than once, I think). $\endgroup$– Włodzimierz HolsztyńskiFeb 7, 2015 at 17:38

$\begingroup$ @WłodzimierzHolsztyński: I believe that the question was asked before the creation of MSE. Back then, mathematical questions of all sorts were allowed on MO. I have voted to close it now, though. $\endgroup$– Alex M.Nov 24, 2021 at 14:00
19 Answers
Polya's "How to Solve It" is a good one. When prepping for the Putnam, I used "Problem Solving Through Problems"
I enjoyed The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics by Shklarsky, Chentzov and Yaglom.
For a slightly annotated list of some books on problemsolving, see https://web.archive.org/web/20170329183237/http://math.mit.edu/~rstan/refs.pdf.
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.

$\begingroup$ The link in the post seems to be dead, but it is still accessible in the Wayback Machine. And various information about this book can be found elsewhere. $\endgroup$ Nov 26, 2021 at 10:46
Knuth's Concrete Mathematics is not only a fun place to learn great combinatorics; it also contains lots of amazing problems.
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. The Art of Problem Solving also publishes special textbooks geared towards olympiad students such as this one on Precalculus.
High school level: Hungarian Problem Book I, II, III, IV
University level: Contests in Higher Mathematics: Miklos Schweitzer Competitions, 19621991

$\begingroup$ Did you participate in the M. Schweitzer Competition once? $\endgroup$ Nov 25, 2021 at 17:58
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.
See also Solving mathematical problems: a personal perspective by Terence Tao.
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.
From a review for Polya's book on Amazon, the books to be read in sequence:
 Mathematical Problem Solving by Alan Schoenfeld
 Thinking Mathematically by J. Mason et al.
 The Art and Craft of Problem Solving by Paul Zeitz
 Problem Solving Strategies by Arthur Engel
 Mathematical Olympiad Challenges by Titu Andreescu
 Problem Solving Through Problems by Loren Larson
Full text of the review below:
By Abhi:
Good aspects of this book have been said by most of the other reviewers. The main problem with such books is that for slightly experienced problem solvers, this book probably does not provide a whole lot of information as to what needs to be done to get better. For instance, for a kid who is in 10th grade struggling with math, this is a very good book. For a kid who is in his 11th grade trying for math Olympiad or for people looking at Putnam, this book won't provide much help.
Most people simply say that "practice makes perfect". When it comes to contest level problems, it is not as simple as that. There are experienced trainers like Professor Titu Andreescu who spend a lot of time training kids to get better. There is lot more to it than simply trying out tough problems.
The most common situation occurs when you encounter extremely tough questions like the Olympiad ones. Most people simply sit and stare at the problem and don't go beyond that. Even the kids who are extremely fast with 10th grade math miserably fail. Why?
The ONE book which explains this is titled "Mathematical Problem Solving" written by Professor Alan Schoenfeld. It is simply amazing. A must buy. In case you have ever wondered why, in spite of being lightning fast in solving textbook exercises in the 10th and 11th grade, you fail in being able to solve even a single problem from the IMO, you have to read this book. I am surprised to see Polya's book getting mentioned so very often bu nobody ever mentions Schoenfeld's book. It is a must read book for ANY math enthusiast and the math majors.
After reading this book, you will possibly get a picture as to what is involved in solving higher level math problems especially the psychology of it. You need to know that as psychology is one of the greatest hurdles to over when it comes to solving contest problems. Then you move on to "Thinking Mathematically" written by J. Mason et al. It has problems which are only few times too hard but most of the times, have just enough "toughness" for the author to make the point ONLY IF THE STUDENT TRIES THEM OUT.
The next level would be Paul Zeitz's The Art and Craft of Problem Solving. This book also explains the mindset needed for solving problems of the Olympiad kind. At this point, you will probably realize what ExACTLY it means when others say that "problem solving is all about practice". All the while you would be thinking "practice what? I simply cannot make the first move successfully and how can I practice when I can't even solve one problem even when I tried for like a month". It is problem solving and not research in math that you are trying to do. You will probably get a better picture after going through the above three books.
Finally, you can move on to Arthur Engel's Problem Solving Strategies and Titu Andreescu's Mathematical Olympiad Challenges if you managed to get to this point. There is also problem solving through problems by Loren Larson. These are helpful only if you could solve Paul Zeitz's book successfully.
To conclude, if you are looking for guidance at the level of math Olympiad, look for other books. This book won't be of much assistance. On the other hand, if you are simply trying to get better at grade school math, this book will be very useful.
"Number Theory: structures, examples and problems", by Titu Andreescu and Dorin Andrica, contains many problems taken from the IMOs.
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
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.
These are some of the books / links which I would recommend:
Functional Equations and How to solve them by Christopher G. Small. This book especially discusses techniques for solving functional equations which appear in the Olympiads.
Geometry Unbound by Kiran Kedlaya.
The Math problems notebook by Louis Funar and Valentin Boju.
Komal, I think is a Hungarian Magazine which contains Olympiad level problems. The archived set of problems along with their solutions can be found at this link.
International Mathematics Competition for University students has problems more or less like the Putnam.
VojtechJarnik is again a Undergraduate Mathematical Competition whose archived problems and solutions can be found at this link.
Problems in Elementary Number Theory by Hojoo Lee and Peter Vandendriessche has nice collection of problems in Number Theory.
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
I prepared my Mathematical Olympiad with "The Mathematical Olympiad Handbook" by A. Gardiner.
Hardy and Williams have "The Red Book of Mathematical Problems" and "The Green Book of Mathematical Problems", with some good analysis, series, combinatorics, 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.
Aiming to add to Pólya's content, I wrote a book about a technique to solve any problem in STEM. It's titled "The TopDown Approach to Problem Solving", ISBN 9798464073296. You can find it on amazon. I hope this is a good fit for you.