On this one, I thought it would be fun to explain how I solved it.
I considered set play (pretend White 'passes' on his first move, and fulfill the stipulation for as many Black defenses as possible).
1...Kd7 2.Qg4+ Kc6 [2...Kd8 3.Rf8#; 2...Ke8 3.Qc8#] 3.Qc8#
In other words, I don't need to take any special measures against 1...Kd7; I just ensure my Q has access to the c3-h8 diagonal. This frees me to devote all resources to defeating 1...Kb7.
If Black is allowed to play 1...Kb7 and 2...Ka7, White has difficulty constructing a mate. I want to play something like 2.Rf7+ to keep him out of a7. After this check, he either returns to c6 for 3.Rc7#, or gets nailed to an edge (2...Kb8 or 2...Ka6). To give mate against both of those, my Q must already be on a square that connects to both the 8th rank and the a-file. There is only one such square.
1.Qd4!! A great key move. Without the above logic, a very hard first move to find. In fact, all the defenses but one are known already. The only one left to consider is the acceptance of the sacrifice, 1...cxd4 2.Rf7! Now possible because the stalemate is relieved. 2...Kc5 (or 2...d3) 3.Rc7#