Mate in 1

Well, since the side giving mate is not specified, I assume it's up to us to figure this out. If white were to move, it would be something like Re1. But if it is white to move, then previous move was made by black and there's clearly no move that could possibly have been made. Plainly, if it's white to move, then it's an impossible position.
Therefore, the answer is Bg2#

On the other hand, the position in any given problem doesn't HAVE to be legally achievable, so I'd say you should've provided more conditions.

Right. The convention in problem chess is always: White to move except he can't.

Btw, illegal problems are worthless. No expert looks at such junk.

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