Is there an index for solutions to American Mathematical Monthly problems?
This is somewhat redundant with Igor Pak's P.S., but I found an index for the 1918-1950 range in The Otto Dunkel memorial problem book, starting on page 80 of this pdf.
I found more: Stanley Rabinowitz and others had the same idea a couple of decades ago, and they started compiling indices, not just for the Monthly. Here are Google Books and official links for the two volumes I found:
1975-1979 (published 1999) Website
1980-1984 (published 1992) Website, Errata
Googling "Stanley Rabinowitz 1985-1989" led me to this:
The labor involved in classifying and indexing the 5-year indexes has proved to be far greater than anticipated. We hope to have the 1975-1979 Index out later this year and the 1985-1989 Index is well under way. [Stanley Rabinowitz, June 1996]
I don't know if the 1985-1989 index was ever finished, but presumably there's at least a partial draft out there somewhere. The website http://www.mathpropress.com/ of Rabinowitz's company includes contact information. They say they're looking for authors to publish, among other things, "indexes to mathematical problems or results", but the page apparently hasn't been updated since 2006.
Tim, you are right - this setting they have is a bit unfortunate for such a great treasure. However, in practice searching for the problem's name in GoogleScholar works fine (use advanced search to restrict your search to the journal, etc.), so I am not sure there is a great need to compile the master index. BTW, other journals have also many interesting problems, notably SIAM Review (I especially like Problem 66-11 which now has its own Wikipedia page).
About problems with/without solutions. There is quite a bit of neglect here. As I explain here, Dyson published his famous "partition rank conjecture" as a Monthly problem, and that problem motivated Fine to prove his famous identities. The problem was solved by Atkin and Swinnerton-Dyer about 10 years later, but you will never hear of this on the Monthly pages, I believe.
In another interesting example, Problem 4325 is a classical "inscribed square" problem (see here and here). Even though it has a "solution", it is important not to take it too seriously.
P.S. One bonus comment: there is a Russian book of Monthly problems, from 1918 to 1950, all with solutions copied and translated from English. If your problem is this old, you can look up the date of the solution in there (and then read it in English).
UPDATE Another example just occurred to me: in this nice paper the authors explain how Erdős posed Problem 3763 (on convexifying polygons), how Sz.-Nagy's solution was incorrect, and how a generation of researchers tried (and, occasionally, succeeded) fixing this solution. The point is that it's basically impossible to do this kind of followup on every Monthly problem.
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.)