The answer to the given question above would be the first option. The sentence that corrects the fragment by including it in the subject is the following one: Cities in many areas of the world have hosted the Olympics.
The PP (prepositional phrase) "in many areas of the world" specifies which kind of cities have hosted the Olympics and therefore includes the fragment in the subject. In the second sentence, "have you lived in many areas of the world", functions as the PP complement of the verb "to live". Finally, in the last example, "in many aeras of the world", the PP functions as a modifier of location of the main sentence.
