die Olympiade
Olympics
In general usage, a term for the Olympic Games, that is, the major sporting event itself.
Die nächste Olympiade der Sommerspiele findet in Paris statt.
The next Summer Olympics will take place in Paris.

Olympics / Olympiad
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Olympics
In general usage, a term for the Olympic Games, that is, the major sporting event itself.
Die nächste Olympiade der Sommerspiele findet in Paris statt.
The next Summer Olympics will take place in Paris.

Although 'Olympiade' (Olympiad) originally referred to the four-year period between the Games, today in general usage the word is used almost exclusively for the Olympic Games themselves. The original meaning is only relevant in historical contexts now.
Die ganze Welt schaut bei der Olympiade zu.
The whole world watches the Olympics.
In the sporting context, 'Olympiade' (Olympics) is often used with the verbs 'stattfinden' (to take place), 'gewinnen' (to win), or the construction 'an der Olympiade teilnehmen' (to take part in the Olympics).
Die Athletin möchte unbedingt an der nächsten Olympiade teilnehmen.
The athlete really wants to take part in the next Olympics.
You can recognize this meaning from the fact that 'Olympiade' (Olympiad) is almost always combined with a subject area. This creates fixed terms such as 'Mathematik-Olympiade' (Mathematics Olympiad), 'Physik-Olympiade' (Physics Olympiad), or 'Schach-Olympiade' (Chess Olympiad).
Er bereitet sich seit Monaten auf die Chemie-Olympiade vor.
He has been preparing for the Chemistry Olympiad for months.
competition / contest
'der Wettbewerb' and 'die Olympiade' can both mean that several people compete against each other. 'Wettbewerb' is the general term for many kinds of competition, for example at school, in sports, or for a prize. 'Olympiade' is more specific: in everyday language, it often means the Olympic Games or a subject-based competition, for example 'eine Mathematik-Olympiade' (a Mathematics Olympiad). The words can be confused because an Olympiad is also a competition. But not every competition is an Olympiad. 'Olympiade' sounds more natural when the competition has this special name or this large, official form.

Wir nehmen morgen an der Mathe-Olympiade teil, weil unsere Lehrerin uns angemeldet hat.
We are taking part in the Math Olympiad tomorrow because our teacher signed us up.
Der Wettbewerb endet am Freitag, und danach wählt die Jury die besten Fotos aus.
The contest ends on Friday, and after that the jury will choose the best photos.