We recommend a very useful set of lecture notes for preparing for the Mathematics Olympiads: the Mathematics Notebook published by the Subalpine Mathesis Association – Bettazzi Section.
- Olympiad Mathematics Notes ▸ A. Astolfi, G. Audrito, A. Carigano, F. Tanturri
(second edition 2010, revised and expanded by G. Audrito, G. Distefano, R. Maffucci, L. Prelli, F. Roman)
https://www.mat.uniroma1.it/sites/default/files/liceomatematico/olimpiadi/dispenseolimpionichetorino.pdf
This text was designed as support material for students who want to prepare for the Mathematics Championships, providing a solid theoretical foundation before tackling more advanced problems.
As the authors write:
We tried to address this book to the widest possible audience of students, maintaining a rigorous approach while avoiding unnecessary formalism and emphasizing practical aspects.
These notes are very clear and accessible, making them particularly suitable also for students studying independently.
For many topics, they can represent an excellent starting point for beginning the study of olympiad mathematics.