Fellow student asked question from take-home exam on Stack Exchange

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 fellow student who cheated?", to which the answer is "yes", as discussed endlessly on many previous questions on this site, e.g.:

Is it okay to report classmates cheating on exams?

Is it reasonable to report another student for cheating when it has no impact on me?

When is it acceptable to report classmates who cheat on an exam?

Should I report cheating to my professor? If so, how?

One can debate (as some here are doing) the viability of take-home exams as an assessment tool, since such exams create relatively easy opportunities for cheating, but that is besides the point. The student cheated, and should be reported. The "moral dilemma" you are referring to is a completely bogus one created by the outdated cultural conditioning that exists in many societies according to which "ratting people out" for bad behavior is morally wrong -- see my analysis of this issue in this answer.

Edit: I noticed that you wrote in a comment that “his account is filled with questions related to assignments in different courses”. Thus, he is not just a cheater, he is a serial cheater, which makes the case for reporting him much stronger. I suggest saving evidence of his cheating (screenshots etc) before reporting to prevent him from hiding his wrongdoing by deleting the questions.

If you are confident that the instructor will be discreet about the identity of the informant (you), then it can't hurt to let the instructor know.

But your concern seems to be more about the exam design and the course grading scheme than about whether your classmate played fair or not.

Therefore I'd suggest that you inform the director of graduate studies in your department instead. I'd usually feel more confident about someone at that level being discreet, and also that's the person who could provide support to the instructor for improving the exam design and the grading scheme in future semesters.

I'd suggest making an appointment to tell the administrator in person rather than sending an email. (You can request the appointment either with the secretary or by emailing the administrator. If anyone asks you what it's about, say it's a confidential matter.)

The meeting can be brief, but do close the door.

You have the third option, one of anonymous reporting. Create a brand new account, like z54nk7 at gmail and use it solely for this job. Use a public network, like Starbucks while reporting. You are not breaking any law, you only make sure your anonymity is granted. While this is less credible, it will surely trigger the curiosity of the instructor, which will compare the received homework with the question on stack exchange.

You insulate yourself from possible retaliation (it may happen that the student in cause is the nephew of the Provost for example) while you still keep your odds of getting a good grade because of distribution bending of grades. Depending on the extent of the inquiry the student may even be expelled, given that all his/her homework will be under scrutiny. It may be also that nothing will happen, given that the instructor is busy with other things (for example, own business or he/she is overwhelmed with other classes/homework).