Besides the trivial case where you later find the answer by yourself, when would one answer their own question?
For example, if I found a cool trick while going through some math contests, could I share them in the form of a question and answer?
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1$\begingroup$ Consider using a blog or personal website instead for such things. Questions which make it seem as if you are talking to yourself by asking a question for which you already know the answer, rather than asking for someone else to answer, are not the intended purpose of this site. As the "tour" says, this site is about getting answers - it's intended for questions to which you don't know the answer. At the very least, you should clearly indicate in the question that you are not actually looking for an answer, so people don't waste time trying to answer, and so they can vote accordindly $\endgroup$– Carl MummertJun 28, 2016 at 2:09
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$\begingroup$ See also meta.math.stackexchange.com/questions/1190/… $\endgroup$– GEdgarJun 28, 2016 at 13:31
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In general I don't think it should matter who answers a question. If a question meets the standards for the site, you can answer it or someone else can answer it. So I would say that you can selfanswer any question that would otherwise be acceptable for the site.
Now, others might like you to show a bit of restraint and not post routine exercises about trivial facts. Maybe you could reserve answering your own questions to when you have that "really nice question". So, if you "found a cool trick while going through some math contests" and wanted to share this, that would be fine.
Please remember to search for duplicates before posting. It would be annoying (to you) if your question was closed because it is a duplicate.
Also, some have suggested to mention in question itself that you intend to answer the question yourself.
Oh, and this has been discussed in the past several times:
