Problem books in higher mathematics
Solution 1:
Try Berkeley Problems in Mathematics, a collection of Berkeley's preliminary exams for graduate students.
Solution 2:
This is a problem book for complex analysis, and I know there is one for real analysis, titled A Problem Book in Real Analysis. I'm fairly certain that parallel ones exist for Introductory Abstract Algebra
Solution 3:
Try "Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991" - it's a book that contains problems (and their solutions) from the hardest higher mathematics competition (harder than the Putnam and the IMC). The contestants are given a week to solve the problems, and they can consult literature.
The topics "range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, topology, to set theory". Beautiful and engaging problems, written by Hungrian mathematicians, some of them well-known.
Solution 4:
I don't know if this is the type of thing you are looking for, but T. Y. Lam has a problem book associated with his A First Course in Noncommutative Rings, which contains all the solutions to the problems in the "First Course."
Solution 5:
I am very fond of the book 1001 Problems in Classical Number Theory by Jean-Marie De Koninck and Armel Mercier. As suggested by the title, the book consists of 1001 problems in number theory, broken up into the following sections:
- Mathematical Induction and Combinatorics
- Divisibility, Prime Numbers
- Representation of Numbers
- Congruences
- Primality Tests and Factorization Algorithms
- Integer Parts
- Arithmetical Functions
- Solving Equations Involving Arithmetical Functions
- Diophantine Equations
- Quadratic Reciprocity
- Continues Fractions
- Classification of Real Numbers
The book has more than 200 pages of solutions, which often times provide a rough outline rather than a full solution, giving one a path to follow in order to fill in the details oneself.