This is the 2104th most frequent German word.
zuständige
"Zuständige" means "responsible" or "competent" in the context of having authority or being in charge of something.
In this sentence, 'zuständige' is used as an adjective to describe the office that is in charge of handling the applications.
Das zuständige Amt bearbeitet die Anträge.
The responsible office processes the applications.
Here, 'zuständige' acts as an adjective describing the person who holds responsibility for the given task.
Wer ist hier die zuständige Person für diese Aufgabe?
Who is the responsible person for this task here?
'Zuständigen' is again used as an adjective, referring to the authorities that have the appropriate jurisdiction for the examination.