[FEN "5K2/k7/4P1p1/8/8/8/4b3/8 w - - 0 1"]
A friend of mine is challenging me to find a solution. It’s White to play and DRAW.
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White is one tempo short of catching the pawn - if White could make two moves immediately it would be a draw as white would just take the black pawn. But they can't, so white has to find a threat which black has to respond to which gains them that move. The only threat they can make is to queen their pawn, and apparently black can stop that with their bishop by moving it to b5. But ... if white can combine moving their king toward the queening square with attacking the bishop, and hence threatening to queen the white pawn, they can gain the tempo. This suggests moving the king towards the queening square via c5, because the king on c5 can attack a bishop on b5 that is stopping the white pawn queen. At that point Black will have to move their bishop or protect it, and white gains the tempo they need. Putting this together leads us to the following - note the move order also means that the black pawn interferes with the Black Bishop's movement, stopping it get to h5 which would be the other square it could use to stop the white pawn queening
[Result "1/2-1/2"] [FEN "5K2/k7/4P1p1/8/8/8/4b3/8 w - - 0 1"] 1.Ke7 g5 2.Kd6 g4 3.e7 Bb5 4.Kc5 Ka6 5.Kd4 g3 6.Ke3 g2 7.Kf2 Bc6 8.e8=Q Bxe8 9.Kxg2 1/2-1/2
If you like this kind of thing have a look at The Reti study which is a more famous relation of this problem. It also turns out that this problem was composed by Reti himself.