The Arab professor, should be accepted in my opinion. How would we say, The smart professor? Is this not the same as The professor is smart?
structure-wise, they are not the same actually.
When you use the verb (to be) in a sentence, like "the professor IS smart" you are telling a fact. In this instance, the adjective "smart" is predicative (i.e. used to tell something).
However, in the phrase (the smart professor), we are dealing with a noun and its adjective (adjectives in Arabic come after the noun and mimics its status). In this instance, the adjective is called "Attributive," that is something attached to the professor himself. We are not about telling something here, we are simply stating the professor as it is and attaching an adjective to it, and the sentence is yet to be completed by other elements to understand the whole meaning.
I wrote "the professor is arabian" and was marked wrong. I know not all arabs are Saudi Arabians, but I thought that all arabs were arabian...
I think an individual is called "Arab" rather than "Arabian" - The same way you might say Swedish, but Swede for the individual, or English but Englishman for the individual.
Though I think they should have added (a) before (Arab).