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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
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    $\begingroup$ 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. $\endgroup$ – user37 Apr 21 '14 at 17:40
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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.
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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.

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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.
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  • $\begingroup$ +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. $\endgroup$ – Benjamin Dickman Dec 16 '17 at 19:14

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