I am an Undergraduate student ready to enter into Graduate School. I will be studying Mathematics. In my Undergraduate studies, I came across various Problem books and followed some of them. The results have been outstanding. I wanted to know if there are any more problem books which I missed or should follow apart from the ones listed below:

1. Problems in Mathematical Analysis by Kaczor & Nowak, Vol 1,2,3
2. Berkeley Problems in Mathematics
3. Challenging Problems in Linear Algebra by Fuzhen Zhang
• I'd like to suggest more than just problem books. Many incredible textbooks have incredible exercises; Jacobson is one of my favorites for introductory algebra, and certainly many others have exercises as essential to the learning process. Atiyah-Macdonald and Hartshorne are the first that come to mind. Hatcher too, for algebraic topology. – user37 Apr 21 '14 at 17:40

There are certainly many more. The following list is not meant to be exhaustive but meant just to give you a selection of the many more books that you have missed:

1. Halmos: Linear Algebra Problem Book, A Hilbert Space Problem Book.
2. George Polya and Gabor Szegö: Problems and Theorems in Analysis - I, II.
3. Ram Murty: Problems in Analytic Number Theory.
4. Jody Esmonde, Ram Murty: Problems in Algebraic Number Theory.
5. Gelbaum: Problems in Analysis, Problems in Real and Complex Analysis.
6. D. Aliprantis and O. Burkinshaw, Problems in Real Analysis.

Following up on Mike Miller's comment that "many incredible textbooks have incredible exercises," you might want to take a look at Pugh's Real Mathematical Analysis for its excellent selection of problems.

I would also also like to recommend Steele's The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities as a book both entertaining and instructive in inequalities and problem-solving techniques.

I like these books:

1. Putnam and Beyond by Gelca, and Andreescu. Brief introductions to mathematical topics from all over the curriculum and a number of competition-style problems to try.
2. A Friendly Introduction to Number Theory by Silverman. The open-ended way in which the problems in this textbook are phrased, are suggestive of the kinds of questions that a researcher might has.
3. Discrete and Combinatorial Mathematics by Grimaldi. This is a textbook that must have more than 1000 problems, many of which are quite good, challenging and give students access to a more general theory than what is being covered in the book.
• +1 for "Putnam and Beyond" alone, which is available in full after a single google: link (pdf 815 pages).  I mentioned the book in an earlier answer to MESE 10556. – Benjamin Dickman Dec 16 '17 at 19:14