Formal:
* La profesora me entrega el examen. (The teacher gives me the exam.)
* La profesora me proporciona el examen. (The teacher provides me with the exam.)
* La profesora me administra el examen. (The teacher administers the exam.)
Informal:
* La profesora me pasa el examen. (The teacher passes me the exam.)
* La profesora me da el examen. (The teacher gives me the exam.)
* La profesora me entrega el examen. (The teacher hands me the exam.)
The best choice depends on the context and your desired level of formality.