These are some of the books / links which I would recommend:
Functional Equations and How to solve them by Christopher G. Small. This book especially discusses techniques for solving functional equations which appear in the Olympiads.
Geometry Unbound by Kiran Kedlaya.
The Math problems notebook by Louis Funar and Valentin Boju.
Komal, I think is a Hungarian Magazine which contains Olympiad level problems. The archived set of problems along with their solutions can be found at this link.
International Mathematics Competition for University students has problems more or less like the Putnam.
Vojtech-Jarnik is again a Undergraduate Mathematical Competition whose archived problems and solutions can be found at this link.
Problems in Elementary Number Theory by Hojoo Lee and Peter Vandendriessche has nice collection of problems in Number Theory.
