Putnam preparation

The best preparation for the Putnam competition is going over the problems from previous years. Problems (with solutions) can be found on Kiran Kedlaya's archive.

There are also four published books, containing all past problems up to 2016. In the books the problems are discussed in detail, and sometimes given multiple solutions:

The William Lowell Putnam Mathematical Competition: Problems and Solutions: 1938-1964, by A.M.Gleason, R.E.Greenwood, L.M.Kelly, MAA (1980).
The William Lowell Putnam Mathematical Competition : Problems and Solutions 1965-1984 (MAA Problem Book Series) by Gerald L. Alexanderson, MAA (1985).
The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary (MAA Problem Book Series) by Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil, MAA (2002).
The William Lowell Putnam Mathematical Competition 2001-2016: Problems, Solutions, and Commentary (MAA Problem Book Series) by Kiran S. Kedlaya, Daniel M. Kane, Jonathan M. Kane and Evan M. O'Dorney, MAA (2020).

Other recommended books are:

Problem solving through problems, by Loren Larson. Springer (1992).
Putnam and beyond, by Titu Andreescu and Razvan Gelca. Springer (2007).
Problem-Solving Strategies, by Arthur Engel. Springer (1998).
A problem seminar, by Donald J. Newman. Springer (1982).
The Art and Craft of Problem Solving, by Paul Zeitz. Wiley (2006)
The Mathematical Olympiad Handbook, by A. Gardiner. Oxford University Press (1997)
Winning solutions, by E. Lozansky and C. Rousseau. Springer (1996)
Problems from the book, by Titu Andreescu and Gabriel Dospinescu. XYZ Press (2010).
Problems in real analysis: advanced calculus on the real axis, by Teodora-Liliana Radulescu, Vicentiu D. Radulescu and Titu Andreescu. Springer (2009).
Contests in higher mathematics. Miklos Schweitzer Competitions 1962--1991. Edited by Gabor J. Szekely. Problem Books in Mathematics. Springer-Verlag, New York (1996).
Mathematical morsels by Ross Honsberger. The Mathematical Association of America (1978).
The USSR Olympiad problem book by D. O. Shklarsky, N. N. Chentzov and I. M. Yaglom. W. H. Freeman and Company, San Francisco and London (1962).
Generatingfunctionology, by Herbert S. Wilf, available online. A. K. Peters (2006).
102 Combinatorial Problems, by Titu Andreescu, Zuming Feng. Birkhauser, Boston (2002).
104 Number Theory Problems: From the Training of the USA IMO Team, by Titu Andreescu et al. Birkhauser, Boston (2006).

Other college-level competitions: