Years ago, I participated in one Mathematical Olympiad in my country (with very good result).
I bought a book with solved problems in this kind of contests. I tried to solve myself about 1 or 2 hours. After that, I read the book and compared solutions. Many times I didn't know anything to do, so reading the book I tried to learn many kind of strategies that are useful in diferent areas (Inequalities are essential in these sort of exams and also some common strategies, as induction, or reductio ad absurdum). By doing that, I had something to try when I didn't know how to continue.
Good luck and don't try to learn solutiones by heart, only strategies.