At https://www.maa.org/math-competitions/putnam-competition it says, "The Putnam Competition covers a range of material in undergraduate mathematics, including elementary concepts from group theory, set theory, graph theory, lattice theory, and number theory," so all you have to do is trundle off to the library/bookstore to find books that cover those topics. Although you also need differential and integral calculus, and linear algebra – I guess the folks who wrote that paragraph thought those topics were too obvious to need mention.
It also says, "Below are some books available for purchase that may help students prepare for this exam:
"The William Lowell Putnam Mathematical Competition Problems and Solutions 1938-1964 A. M. Gleason, R. E. Greenwood, and L. M. Kelly, Editors
"The William Lowell Putnam Mathematical Competition Problems and Solutions 1965-1984 Gerald L. Alexanderson, Leonard F. Klosinski, and Loren C. Larson, Editors
"The William Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary Kiran S. Kedlaya, Bjorn Poonen, and Ravi Vakil"