That's a retro-analytical problem designed to trap even experts.
For non-experts, first note the rule regarding potential en passant captures in a problem diagram. The rule is that such an e.p. capture is illegal unless it can be proved – by a type of backward reasoning called retro-analysis – that the previous move must have been a double-step by the pawn to be captured. See my blog for an example.
The OP problem has two potential solutions: 1.cxd6 e.p. mate which is allowed only IF the last move was necessarily ...d7-d5, and 1.fxg6 e.p. mate which is allowed only IF the last move was necessarily ...g7-g5. We can rule out alternative last moves by Black through a process of elimination. Black's last move couldn't have been ...Nc4-a5, ...d6-d5, or ...f6xg5 because these would imply Black was checking White while it was Black's turn – an impossibility.
What about ...e6xd5, ...g6-g5 ...h5xg4, ...Kd7-e7, ...Kd8-e7, or ...Ke8-e7, where the BK could also have just captured something on e7? If you can demonstrate why each of these is also an impossible last move, well done!
Finally, you are left with ...d7-d5 and ...g7-g5. Are both possible last moves or just one of them, meaning is the solution 1.cxd6 e.p. or 1.fxg6 e.p. mate? That is the tricky question that even experts can get wrong!
In this position a mate in 1 is infact possible.
There is no unique solution to this puzzle, BUT the correct answer is either cxd6# or fxg6# (en passant)!
If you take a closer look at the position, Black can only have moved d7-d5 or g7-g5 on his last move, since everything else (as you can check) was illegal for Black.