The American Mathematical Monthly goes back a long way, to a radically different era in mathematical life, and has never quite caught up with the present in terms of making information readily available. Not being a research journal as such, it mostly falls outside the realm of review databases MathSciNet, Zentralblatt. I guess the assumption used to be that you would have a shelf full of back issues in your office for frequent perusal. Also that you were probably involved mainly in undergraduate teaching and relied on the Monthly for your doses of real mathematics and mental stimulus. There is some online publication now, but only for members.
Having written articles and problem solutions that appeared in the Monthly, I am aware of the searching problem. I've lost track of solutions I wrote, for instance, and can probably never expect to locate them. For older main articles, JSTOR does provide some help. But the problem section is more problematic to search over time. (I wonder whether anyone in the MAA leadership is listening to such concerns?) Its a large membership organization but not as useful in the modern era as AMS has tried to be.
[ADDED] The comments being made here are helpful. The Monthly editors running the problem column used to publish periodic listings of recent unsolved problems, by the way. And the annual indexes of the Monthly give a few clues. But more recent online search tools are much better for searching in subject areas and the like. For example: find all articles/problems/solutions about finite groups, Galois theory, infinite series, etc. I recall a number of short papers giving alternate proofs of Wedderburn's theorem on finite division rings, including Herstein's famous elementary (but long and rather opaque) proof. To find these listed together online requires plenty of time and some luck. A big plus of MathSciNet is the expanding capability of its search engines covering the entire history of Math Reviews. Not exhaustive, but it gives citations and other related references. (But that costs big money and mathematical expertise to build and maintain, thus is far from free to the public.)