OK. With the white king on c5.
The last Black move cannot have been b6-b5, because white would have been in check.
c7-c6 and h7-h6 are out because there are white units on c7 and h7.
It may have been Qb2, blocking a check. But where did the black queen come from? Clearly not a3 or b4, as white would have been in check himself. So Qb3-b2 is the only possibility. But this itself is impossible because the white queen has no legal way to play Qc2+ - it can't have come from b2 itself and none of white's pieces can have moved from b2 giving a discovered check. Hence black's last move was not Qb2.
This leaves b7-b5 as being the only possible last move for Black, and then White has axb6#.
Thus the solution is that the White king has to be on c5, not c4, as with it on c4 it is not possible to prove that Black's last move was b7-b5 (and it's quite easy to construct a series of moves where this is not the case).