I want to know if there are books that give problem sets requiring knowledge of two or more branches of mathematics. For example, there could be a problem requiring geometry, set theory, and number theory, and another problem requiring ring theory and calculus. All the problems should be solvable with merely an undergraduate background.
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You should take a look at the William Lowell Putnam Mathematics Competition 1985-2000: Problems, Solutions and Commentary, by Kedlaya, Poonene and Vakil.
The very focus of the Putnam Competition is to test undergraduates knowledge of large fields of mathematics, with a focus on synthesizing ideas in interesting ways. Many of the problems are quite hard, and require a good deal of breadth of knowledge, but they are not unaccessible, and the book gives hints as well as solutions, so that you can practice many types of problems.
A. Thomas Yerger
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Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by John and Barbara Hubbard offer many problems that tie together analysis and linear algebra. The problem sets however do not require as diverse of a knowledge as competition level problems (as stated by Alfred).
recmath
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