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My question is very soft but I think answerable: are the problems posed in American Mathematical Monthly, say, research type of problems? I suppose not since it is certain that these questions have solutions, but the problems can lead to more questions and then we can say it is research, is this correct?

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    $\begingroup$ No, I don't think so, just "more questions" we cannot say that it is research. The questions must be research questions. $\endgroup$ Jan 27, 2019 at 15:20
  • $\begingroup$ The comment by @DietrichBurde is mostly correct, but I have seen research papers arising from problems stated in the Monthly. Not often, but it can happen. (I've seen 2 examples in the last year, research papers on the same topic, each referring to a different but related Monthly problem.) $\endgroup$ Jan 27, 2019 at 15:38

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Your first question is answered by a definite 'No', because submission of a problem must be accompanied by a solution.
See AMM, paragraph "Proposed Problems or Solutions", from where
"We invite readers to submit original problems that will appeal to readers of the Monthly. Problem proposals must be accompanied by solutions and any bibliographical information that will assist in their review. Submitted problems should not be under consideration for publication elsewhere."
is taken. This endorses your assumption.

Your second raised expectation is correct, and there are certainly instances of posed problems including solutions stretching out to new research.

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