1. This is a near miss:

 FEN Viewer

Black wins only by 1...f2 2.Kxf2 h5 - everything else is a draw. But putting the king in the way of the h-pawn allows it to save the f-pawn.

I haven't ruled out the answer being "you can't".

2. It’s fairly clear a pawn must be on the h file since you cannot capture all three. White will need at least 3 moves to capture both pawns, and if black can play h5, h4, h3, the second pawn captured must be on f3 to control h1. Since putting it there doesn’t work, it’s safe to assume that you must somehow prevent/interrupt him from pushing the pawn to h3

3. This is another very near miss

 FEN Viewer

White captures both the g and e pawns but meanwhile black has Kg7-Kg6-Kh5-Kg4 and wins. Could put the king on a8 instead, but then it saves the e-pawn.

4. Solved! Thankyou, that was quite difficult and rewarding.

Hint in white text:

One of my near misses is very near indeed.

6. I've also found a second solution (not a mirror) that is only very slightly different to the first one.

7. I can’t seem to find a second solution, king has only one square that can’t protect e4 and it doesn’t look like the pawns have any alternative.

White to play

https://media.discordapp.net/attachm...768&height=768

By Alexander Petrovich Kuznetsov

Great ending

8. Originally Posted by Zelgiusfan5000
I can’t seem to find a second solution, king has only one square that can’t protect [..] and it doesn’t look like the pawns have any alternative.
Solutions in white text (verified by tablebase):

Black pawns e4, g3, h6 Black king h8. White king can be either g2 or h3

9. Ah right

10. Originally Posted by Kevin Bonham
Solutions in white text (verified by tablebase):

Black pawns e4, g3, h6 Black king h8. White king can be either g2 or h3
Ah I get it now. With the pawn on h6 rather than h7, it matters in the line with white taking the oppostiion on g4-g6, when black has neither the waiting h7-h6 or Kh6. Very nice.

11. White to play

https://media.discordapp.net/attachm...768&height=768

By Vitold Yakimchik

Great study. I’ll start posting hints in white text now: Black’s main idea is to stalemate

12. White to play

https://media.discordapp.net/attachm...768&height=768

By Oleg Pervakov

Can’t think of a good hint for this one. Enjoy

13. White to play

https://media.discordapp.net/attachm...768&height=768

By Nicolaî Kralin

Most moves are relatively easy to find, but some precision is needed for the finishing blow. Ideally, you’d like to play ba5 when black’s king is on c4, there isn’t any other way to win the c pawn. Black’s king also needs to be able to step inside the h pawn’s ‘box’ whenever you play bh2

14. Originally Posted by Zelgiusfan5000
White to play

https://media.discordapp.net/attachm...768&height=768

By Nicolaî Kralin

Most moves are relatively easy to find, but some precision is needed for the finishing blow. Ideally, you’d like to play ba5 when black’s king is on c4, there isn’t any other way to win the c pawn. Black’s king also needs to be able to step inside the h pawn’s ‘box’ whenever you play bh2
8/8/8/1P6/1P5p/k1p1B1pP/2P1K1Pp/8 w - - 0 1
1. Kf1 h1=Q+ 2. Bg1 Ka4 3. b6 Kb5 4. b7 Ka6 5. b8=B Kb5 6. Bd6 Kc4 7. Bc5 Kb5 8. Be7 Kc4 9. Bxh4 Kxb4 10. Be7+ Kc4 11. Bd6 Kd5 12. Bxg3 Kc4 13. Bd6 Kd5 14. Bb4 Kc4 15. Ba5 Kd5 16. Bxc3 Ke4 17. Be1 Kd5 18. Bg3 Kc4 19. Bh2 and the H and C pawns cannot be stopped by the king.