Many retros have the "Mate in one" stipulation. How can this be a real problem?
With such problems, the direct (i.e. forward) stipulation is generally an implicit way of asking for a retrograde analysis of the position. Of course this is only interesting if the actual solution cannot be found without a correct analysis of the past of the position.
This is the case when the solution involves castling or an en passant capture.
But often the trap is that RA will establish that Black has the move. Then the intended direct mate may be performed by Black (1st situation). It may also be performed by White against any Black 1st move (2nd situation). In any case, the real interest of the problem is usually its retro-analytical content, and the mate in one is the icing of the cake. Some problems explicitly use the "Mate in 1 (Who?)" stipulation, to make sure that solvers won't miss the real point.
Because of the trap intended by the composer, it is most appreciated when the direct stipulation is the simplest possible, e.g. "mate in one!" (see also problems without words). That way, lemmings are sure to make a fool of themselves.
Here is a simple example of the 1st situation:
diagrammes 60, 01/1983
15+15. Mate in 1
And a simple example of the 2nd situation:
W. F. von Holzhausen
Akademische Schachblńtter, 1901
5+2. Mate in 1