Let's be clear here. From your description, the student clearly cheated by violating the take-home exam's stated policy that (quoting from your comment on @user2768's answer) "it is prohibited to ask anyone for an answer". Thus, your question reduces to the generic question of "should I report a ...
I agree with everything in this answer, and would add: It's not just that the other student is repeatedly doing something immoral. He's not learning to solve problems on his own or gaining the understanding of code that he would get by doing so. One day he may get a job based on his degree and grades. What's wrong here is not just that this student is being deceitful, and unfair to other students who do do the work, but that he is potentially harming himself, future employers, and their clients. (Lives may depend on it, even, e.g. if he wrote software for self-driving cars.)
Cheating is a culture. If you don't want to belong to a culture where people cheat you, then it is up to you to stop people from cheating. Also, don't be a cheater yourself, in case that is a problem.
One thing to consider though: can you really say the student cheated - as opposed to some student cheated and opened an account under the student's name?
@cbeleites: No, OP can't say that for sure, and doesn't have to. OP is just bringing important evidence to the attention of the professor; it's the professor's job to evaluate it. The professor will also have other evidence: namely, the exams turned in by the students. If the student's exam answers, or for that matter anyone else's, are obviously copied from the SE answer, done.
There's another problem that takes this beyond normal cheating - any student could see the question and answer (obviously the OP did, though they may have avoided reading the answer), and accidentally have it spoiled for them if they haven't solved it yet. They may not even have read it on the test yet, so they wouldn't know to avoid it. Now, it's on them to not only find an answer, but an answer that doesn't appear to have copied from the SO answer. And Heaven help the poor student that accidentally writes an answer similar to the one posted on SO without knowing about it.
@IllusiveBrian the instructor can always contect SE and request email data to see who actually owns the account. Besides, with it being a serial cheater there is a general trend of copied answers that can be used to show who is or who isnt a cheater.
I agree with everything in this answer except the “moral dilemma” bit. Ratting out people for minor (or legalistic) infractions is unethical, and hugely problematic if it becomes the social norm. It’s an instrument of repression, and this is how you get a totalitarian state. We’ve been there — repeatedly! —, it’s not pretty. Don’t underestimate the importance of a social norm against snitching. (But importantly it doesn’t apply in the question’s context, as repeat cheating in the academic context is hardly a minor misdemeanour.)
@TheGreatDuck Here’s hoping that SE would not give out that information to random folks. Besides being completely unethical it’s likely illegal. When SE cooperated with law enforcement in the past (which they have), they presumably did so in the presence of a court order.
@KonradRudolph You can compare the Stack Exchange moderator agreement, which is eminently short and readable. The privacy policy also applies. Basically in this case, SE prohibits itself from sharing personally identifiable information (except that willingly shared by the users) with third parties, and also passes this prohibition on to the moderators across the network. Consequently, absent a court order to do so, SE can't provide the professor with user account details.
@KonradRudolph my other answer that I linked to also refers to the origins of anti-snitching taboos in totalitarian regimes where snitching is indeed used as an instrument of repression, so you are partially correct. However, in the context of the question, and in most contexts where the supposed “moral dilemma” comes up today (at least in advanced democracies), snitching is the opposite of unethical. In particular, cheating on a take-home exam is neither a “minor” nor a “legalistic” infraction, and there is no real dilemma about whether it should be reported or not.
@DanRomik Right, I saw your other answer and, to be frank, it is dangerously ill-informed in its political history. This isn’t something that only happened in medieval evil kingdoms. It happened, repeatedly, during the 20th century to countries in the Western hemisphere (twice in Germany). It was the norm pretty much everywhere in Europe at any time before that. It has a real risk of recurring in modern societies, and there are already nudges in that direction, for instance in the UK (denouncement of anti-Brexit in higher ed), and much worse in the US (e.g. ICE snitches).
@DanRomik … which is why I mentioned that I agree with the rest of your answer but that your take on the ethics on snitching in general is plain wrong, dangerous, and can’t be left unchallenged.
As far as saving the screenshots goes, if he does try to hide it later, he may also tell the institution that the images provided by the OP are fake. In that case, it would be nice if the OP had the assistance of a moderator available, since moderators are often able to see deleted content (and can also talk with someone from the institution in chat). That may be worthy of a question on the meta site.
@KonradRudolph ok, thanks for sharing your interesting take on this issue. I acknowledge that even in modern democracies there are sometimes situations where snitching is harmful and unethical, so in that sense I agree with you, and don’t see what you wrote as inconsistent with my own views, or as convincing evidence that my earlier remarks are “wrong”, “dangerous” or “dangerously ill-informed”. Be that as it may, we are veering into a political discussion that seems clearly (to me) off-topic and with no relevance to OP’s actual question. Thanks again for your contribution to the discussion.
Discussion on answer by Dan Romik: Fe…
Discussion on answer by Dan Romik: Fe…