Full solution and announcement of the winners. :)
The winner of this round is @kajalpats, congrats for being very fast!
The second place goes to @shortcut who found the winning move a little bit later. Congrats, too!
Here the full solution: 1. g6! And now:
a) 1. ... fxg6? 2. Qg7#
b) 1. ... hxg6 2. Rxg6+! fxg6 3. Qg7#
c)
1. ... Qg3 2. Rxf7! Rxf7 3. gxf7+ Kxf7 4. Rd7+ (fastest).
After 4. ... Ke8 5. Rxh7 follows check mate in (at the latest) 6 moves.
After 4. ... Ke6 5. Rxh7 follows check mate in (at the latest) 12 moves.
After 4. ... Kg8 5. Qxc4+ follows check mate in (at the latest) 19 moves (analysis by means of Stockfish)!
RE: Chess problem 39 / Schachaufgabe 39