Math Olympiads
This page is primarily aimed at motivated Czech high school students but it might be interesting to math teachers and parts of it to all math enthusiasts. That being said, here is my take on what you can do if you are a student in the Czech Republic, like math and want to do more of it.
Czech competitions
Matematická olympiáda (MO): *The* math contest. The first round (take home, harder than the subsequent onsite rounds) is due in December/January. The winners of the national round get to represent Czech Republic at the IMO the next summer.
PraSe (MKS): A correspondence seminar, 8 problem sets per school year, two week-long camps for top contestants each Fall + Spring. Other seminars with similar format include KMS, M&M, Brkos, Strom. If you are serious about math, try iKS (or TRiKS to start).
Náboj: An onsite event for school teams of size 5. Registration around February, takes place in March. A similar (online) event is MathRace (November). A similar (international, online) event is PurpleComet (April).
International summer programs
Math Beyond Limits (MBL): 2-week camp in Poland in early September. Free. Recruitment in April.
AwesomeMath Summer Program (AMSP): 3-week camps in the US in June-August. Recruitment in January-April.
PROMYS Europe: 6-week camp in Oxford, UK. Recruitment in January-March.
Online resources
Art of Problem Solving (AoPS): *The* community if you are serious about olympiad math.
(czech) Knihovna PraSete (MKS): A handout on pretty much any math olympiad topic, all difficulty levels. My handouts are here.