I think that "غرفة أستاذ محمد" is sufficiently definite, and should be considered a correct answer.
"غرفة أستاذ محمد" woul mean "Mohamed's professor's room" (i.e. "the room of the professor of Mohamed")
That actually would work in dialect to mean "Professor Mohamed's room." Not in MSA though.
I was just wondering this. As a dialect learner, I wasn't sure if it was the same for MSA