abgegeben
submitted
Describes something that has been handed over to someone, submitted, or left at a certain place.
Die abgegebenen Hausaufgaben liegen auf dem Schreibtisch des Lehrers.
The submitted homework assignments are on the teacher's desk.

submitted / cast
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submitted
Describes something that has been handed over to someone, submitted, or left at a certain place.
Die abgegebenen Hausaufgaben liegen auf dem Schreibtisch des Lehrers.
The submitted homework assignments are on the teacher's desk.

'Abgegeben' (submitted / cast) is the past participle of the separable verb 'abgeben' (to hand in / to submit / to cast). Here it is used like an adjective before a noun and describes something that has already been handed over.
Alle abgegebenen Arbeiten werden bis Freitag korrigiert.
All submitted papers will be corrected by Friday.
In this meaning, 'abgegeben' (submitted) often appears with things that you submit to a person or institution, for example: abgegebene Dokumente (submitted documents), abgegebene Anträge (submitted applications), or ein abgegebenes Paket (a dropped-off package).
Der Stapel mit den abgegebenen Anträgen ist schon sehr hoch.
The stack of submitted applications is already very high.
In the context of elections and votes, 'abgegeben' (cast) is used almost exclusively with the nouns 'Stimme' (vote) or 'Stimmzettel' (ballot).
Die Zahl der abgegebenen Stimmzettel war niedriger als erwartet.
The number of cast ballots was lower than expected.
In sports journalism, 'abgegeben' (taken / made) describes an action that has been carried out. Typical combinations are 'der abgegebene Schuss' (the shot taken) or 'der abgegebene Pass' (the pass made).
Der erste abgegebene Torschuss der Mannschaft war sofort ein Treffer.
The team's first shot on goal was immediately a hit.
to hand over / to vomit
'übergeben' and 'abgegeben' are closely related because both can express that something goes to a person or office. The important difference is the part of speech: 'übergeben' is a verb and describes the action. 'abgegeben' is an adjective or participle and describes the state afterward: something has already been submitted, handed over, or left somewhere. Learners may confuse the words because both occur in situations involving forms, assignments, or objects. For the action, you say 'etwas übergeben' (to hand something over). For the result, you say 'die abgegebene Arbeit' (the submitted assignment) or 'die Arbeit ist abgegeben' (the assignment has been handed in).

Vor dem Wochenende war der abgegebene Antrag schon im Büro angekommen.
Before the weekend, the submitted application had already arrived at the office.
Ben übergibt dem Lehrer die Mappe, weil er am Wochenende arbeiten muss.
Ben hands the folder over to the teacher because he has to work on the weekend.