Irontiger, you found the right move. Black cannot avoid mate if it captures the rook.
Puzzles just for fun!

But how ? My calculation goes 1.Rf7 Bxe3+ 2.Nxe3 Kxf7 3.Qxh7+ Kf8 4.Bxg6 Be8 and I don't see how to avoid giving the perpetual.
EDIT : never mind... 5.Qh8+ Ke7 6.Qg7#

Yeah, you got it. But my opponent was very competitve and found the best possible defence.
That sounds interesting ! Can you please post the game ?
i can undestand tal dit not find Qf5 g6 Qd7!! g+h Ng5 Kg6 Qe6 K+g5 g3! (treats h4 # so ) h4 is forced f4 wins all blacks pieces and the game

#89 : it is easy to see that there is no "mate in one" as no move can control both a2 and b1.
It is also easy to see that the Black king does not belong to where he is : as b2 and c2 have not moved, the king must have gone on a2 from the first row, but there is no way in or out for a bishop on c1. Hence the position looks illegal. (the bishop on a7 puzzled me for a while, but there is a way to put it here via f8-h6-e3).
My best guess is that the board is upside down so that the mating move is 1.b2-b1=Q# (or rather g7-g8=Q#)

For #89 I already gave the co-ordinates so it can't be upside down. Of course, the king also can't be on a2, a1 or b1 as it would be an illegal position (as Irontiger pointed out, but failed to see a very interesting fact).
Whether you say it the genius of the composer or a funny thing about the position. You just put the king to any square (except a2, a1 & b1) to make it a legal position and it's mate in one.
#82 : Rf7 looks tempting, but I do not see an immediate mate or win of material once Black accepts. Perpetual is at hand of course, but that's not enough...