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?
Problems in American Mathematical Monthly are research type
soft-question
Related Solutions
Tristan Needham,
Visual Complex Analysis, Oxford Univ. Press.
"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. ... to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices—but his are interesting." —Ian Stewart
Here's one figure from the book, p.135:
You can almost guess the theorem from the figure: The two spheres $S_1$ and $S_2$ are
orthogonal iff the two circles $C_1$ and $C_2$ are orthogonal.
The n-body problem dates back to the ancient Greeks and was once considered the key to understanding the movement of the planets, and, by extension, the very nature of the Universe.
A "geometrical" (i.e. exact) solution was always hoped for and expected, but the desire to compute orbital motions in practice led to the development of numerical methods; in the meanwhile, it was discovered that the case $n>2$ is chaotic (an early motivation for chaos theory).
Both numerical analysis and chaos theory have since dwarfed the original problem in terms of the amount of research effort dedicated to them, and while it's difficult to say that the n-body problem is "forgotten" (if it were, we couldn't be discussing it) I'm not aware of any substantial theoretical work (let alone progress) done on it for several decades now beyond what is already covered by the more general field of numerical methods.
Add to this the fact that the relativistic version of the n-body problem is fundamentally different from the classical formulation, leading to a solution for the classical problem being of much lesser interest in today's context of large-scale celestial mechanics than it would have been two centuries ago, and you have this once monumental problem being now reduced almost to its historical importance alone.
Best Answer
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.