Most published math papers are answers to open questions posed by the authors of the papers, right? So why is this problem that the paper responds to is never explicit in the text by the authors? Would not that be an important thing since it would save time for other mathematicians not to waste time formulating problems that have already been answered? Or is an expert in the field able to identify the open problem that a particular paper responds to even if it is not explicit in the text? Could someone explain to me in detail how this works?
Most published math papers are answers to open questions posed by the authors of the papers, right?
Most math papers are proofs of theorems. For instance, "I" might prove the following theorem:
Let m>n>0 be integers. Then for a=m^2-n^2, b=2mn, c=m^2+n^2 it holds that a^2+b^2=c^2.
"Open questions" usually refers to problems left open by someone else in another paper (or sometimes, left open by the authors in a previous paper). Now, "me" proving this theorem is probably in response to a question I had: for instance, I might have wondered which integers satisfy a^2+b^2=c^2. However, if no one else had previously considered this problem, it doesn't count as an "open problem".
Typically, in the conclusions sections of a paper, the authors might mention several problems that they consider interesting but didn't manage to solve. For instance, in my paper I might mention the question:
Are there any strictly positive integers satisfying a^3+b^3=c^3?
If someone subsequently were to solve this problem (whether in the margin of their paper or not) they would almost certainly write something to the effect of:
We solve an open problem due to van der Zanden , showing that there are no integers satisfying [...].
The point of mentioning that a question is "open" is to:
Credit the ones that formulated the problem (formulating a "good" problem can be as hard as solving it).
Demonstrate the significance of the problem (they considered it interesting enough to mention as an open problem).
Show that the problem is not trivial (they tried solving it but couldn't).
If I am solving an "open question" that I posed myself then none of these points apply (I could prove an utterly trivial, uninteresting theorem and then invent an equally uninteresting "open question" question that it answers) and so it doesn't make much sense to state an "open question" if there was no open question to begin with. Sometimes people answer their own open questions, but only if they previously stated them in another paper. If I just prove an interesting theorem (e.g.: "All swans are white") then reformulating the theorem as a question (e.g.: "We answer the question of whether all swans are white") doesn't add anything to the paper - especially if no one previously considered that question.
Note that in many math papers, the authors also invent the problem they solve (most math papers solve problems that are not well-known, even in the same community). Some problems require quite a bit background to formulate, in particular if new terminology is invented. Thus, the problem cannot be formulated in the abstract in detail. Besides, many journals do not allow/dislike math formulas in the abstract.
Another point is the following: Solving open problems is not important - solving important problem is. Thus, one actually needs to sell the problem, motivate why a solution is needed, perhaps give historical background and applications. Also solving old problems with new techniques, or make unifying proofs (putting same phenomena in different guises in one general form) is also something that happens. For new techniques, see the history of the prime number theorem, and for a unifying definition, see "cyclic sieving phenomena" by Reiner and Stanton.