On the case presented I have to say, the professor is being unethical. Looking up questions and answers to previous exams is practically studying. The point is to know the subject matter not to read it through a specific source or in a specific format.

He should stop being lazy and vary the questions every year. If after a few years there is a body of previous exams covering the entire subject matter, then great the students "cheating" will be studying the entire course in Q/A format.