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][1].

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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][2] (published 1999) [Website][3]

[1980-1984][4] (published 1992) [Website][5], [Errata][6]

Googling "Stanley Rabinowitz 1985-1989" led me to [this][7]:

> 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.


  [1]: https://docs.google.com/leaf?id=0B_pEp00B111JNjZjZjE1NGUtN2YwOC00OTUyLWE4NmYtNzEzY2VhMzk5Y2Iz&hl=en
  [2]: http://books.google.com/books?id=KX6D6hefyA0C&lpg=PP1&dq=%2522index%2520to%2520mathematical%2520problems&lr&pg=PP1#v=onepage&q&f=false
  [3]: http://www.mathpropress.com/books/1975index/
  [4]: http://books.google.com/books?id=n-F52zK2UAgC&lpg=PR7&ots=cwyXL3xTg4&dq=%2522mathematical%2520monthly%2522%2520problems%2520solutions%2520index&lr&pg=PP1#v=onepage&q&f=false
  [5]: http://www.mathpropress.com/books/1980index/
  [6]: http://www.mathpropress.com/errata/1980errata/errata.html
  [7]: http://books.google.com/books?id=RJVii6vc0NAC&lpg=PP11&ots=C4dnFu344h&dq=Stanley%2520Rabinowitz%25201985-1989&pg=PP11#v=onepage&q=Stanley%2520Rabinowitz%25201985-1989&f=false