As the title indicates, I'm looking for a few suggestions on problem books. The problems should be a bit less demanding than Putnam problems. Like the Putnam, however, the prerequisites should be minimal; some basic abstract algebra, real analysis, and number theory. (Actually basic problems in these fields would be great. For example, I've seen Putnam problems using only the idea of a commutative & associative binary operation. Cf. 2012 Problem A2). Please, I would love a few suggestions! An additional suggestion for a book with no prerequisite, i.e. for high school students, would also be great.
I heard the Art of Problem Solving site has published dozen of problem books in number theory, abstract algebra, inequalities, intro to real analysis, complex variables. You can take a look at their site and choose the books you want to buy. The books are for those HS students who are interested in participating in math olympiad events. But that is a short term goal because it is for undergraduates. If you intend to go to graduate school, then the better option for you would be to start reading graduate level books as soon as you are able to.