Pinning down the notion of an "open problem" seems to be… an open problem in itself, so much does the definition vary depending on who is using it and in what context.
In teaching
In teaching, the term "open problem" is generally used in contrast to an exercise that consists of answering a sequence of guided questions in order to reach a result by following a path set by the teacher. This type of problem is called "closed," as is, generally, any question preceded by the instruction "prove that," which offers no opportunity to explore and formulate conjectures.
This brings us to the other definition of open problems, whose original meaning is far broader. In ancient Greek, problêma is built on the prefix pro, meaning "in front of," and the root ballein, "to throw." Its literal sense is thus "obstacle," and by extension "proposed question," "subject of controversy." It is indeed in this sense that the notion of open problem is understood in mathematical research. There, open problems are problems whose answer is not (yet) known.
Conjectures and millennium problems
