Mathematics Olympiads are mathematics competitions. In some countries, mathematics Olympiads refer to all math competitions, while in some countries, including the United States, math Olympiads refer to proof-based math competitions.
The International Mathematics Olympiad (IMO) is the most famous math Olympiad. In USA a Math Olympiad would be United States of America Mathematical Olympiad that is famous(you'd have to take American Mathematics Competition 10/American Mathematics Competition 12 and American Invitational Mathematics Examination and do well on it to qualify for it). Another in USA is USAMTS which if you do well can qualify for AIME. Both Olympiads are from USA & are proof based.