The basic grammar rule of the question is subject/ verb agreement. Essentially, we want to stress that the if the subject of a sentence is plural, more than one, then the verb must reflect this. For example, "The origins of most sports are unknown." In this sentence, the subject, "origins" is plural. This means that the verb has to be plural, so that the verb will be "are," to reflect the plural of the state of being verb, "is." I think that the sentence is seeking to clearly establish the relationship of the past to the present day, and in doing so, I find that a simple movement of the verb from the singular to the plural is all that is needed. The problem with the other options is that they are either grammatically inaccurate or their verb tense is not consistent with the meaning of the sentence. For example, "The origins of most sports have been unknown" is reflecting an improper verb tense, although its subject/ verb agreement is accurate. It seems to me that we are looking for a correct use of verb tense and correct subject/ verb agreement, which leads me to the third option of the four featured.
the origins of most sports are unknown--is the correct form, following the subject-verb agreement.