It is White's turn to move. How can White avoid a checkmate?
[FEN "8/1kpP2q1/1p2P1p1/1P3pP1/5p1K/5P2/6P1/R7 w - - 0 1"]
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White is clearly in a dire situation, since Black is threatening mate in one on either h7 or h8 if White doesn't do anything. But the White king can't move nor can White get rid of any of the pawns that surround the king. White's rook, the only White piece, is too far away to do anything. The move 1. Ra8? to try and stop 1... Qh8# fails to 1... Qh7#. Promoting White's pawn with 1. d8=Q? also fails to 1... Qh7#. So what can White do now?
Assessment Of Available Options
In order to prevent Black from moving the Black queen, White must give a check to the Black king so that Black is legally obligated to move the king instead of the queen. However, what check is there for White to give? Ra7+ doesn't work because the Black king will just take it and White cannot stop both mating threats. It seems that White is totally losing.
There is only one move that can save White, which often goes unseen by many first-time problem solvers: underpromotion to a knight with check: 1. d8=N+! After 1... Kb8 (not 1... Kc8?, as that allows 2. Ra8#), White can forcibly checkmate Black in 4 moves with a constant stream of checks as any non-checking move loses. Therefore, White is not losing as it seemed before!
[FEN "8/1kpP2q1/1p2P1p1/1P3pP1/5p1K/5P2/6P1/R7 w - - 0 1"] 1. d8=N+ Kb8 2. Nc6+ Kb7 3. Ra7+ Kc8 4. Ra8+ Kb7 5. Rb8#
It is important to note that while 2. Nf7 seems like an option to block the Black queen, it actually loses as Black can then play 2... Qxa1!, and it is checkmate on the next move with 3... Qh1# no matter what White plays.
This is actually a cool position. Although, not easy to see as first sight, white can execute an Arabian Checkmate. Like others mentioned, executing the move
d8=N+ is the only move that saves white from checkmate. This would force a check, but also allow white to win with a maximum of 4 moves proceeding an Arabian Checkmate.
It's an underpromotion problem, with promotion to a knight on d8 with check by 1.d8(N)+ to avoid checkmate by the black queen on the ensuing move on the "h" file. All ensuing white moves must be with check for the same reason. If 1..., Kc8 then 2. Ra8# is mate, so the king must go to b8 with 1..., Kb8 after which White can follow up with 2. Nc6+ Kb7 (or 2..., Kc8 3. Ra8+ Kb7 4. Rb8#) 3. Ra7+ Kc8 4. Ra8+ Kb7 5. Rb8#.