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 links for the two volumes I could find: 

[1975-1979][2] (published 1999)

[1980-1984][3] (published 1992)


  [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://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