I believe we already have a couple of questions that are, while understandable at the high-school or even lower level, have no accepted answer in even research-level Maths.
I'm not referring to in-debate answers (such as "is 0 a natural number?"). I'm talking about problems that have not yet been solved.
Imagine if the question "Are there any integer solutions to a^n + b^n = c^n, where n > 2?" were asked fifty years ago. It's a simple question, but at the time there were no actual answers that the Mathematics community could give, besides, "We don't know."
(Note that, if someone actually were to ask Fermat's Last Theorem, it'd be quite suspicious and probably a trolling attempt. I'm speaking of questions that are not as well-known yet still pose trouble for research-level mathematics today)
Should the answer be "We don't know, and here's why. But we might one day find out." ? (With a decently typed out "why")
Also, what is the duty of the asker to choose an answer to accept? If the best answer is the answer in the form above, should it be accepted? Or should the question be left unresolved?