Black to move. Can black hold this endgame? How should black proceed?
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Don’t have time for the full analysis, but Ke7 immediately suggests itself- this is what I call the 3-squares direct opposition, whereas the normal direct opposition is across one open square between the kings. The principle is the same, but two ranks/files further apart. The pawns are locked unless white pushes d4. So, after 1. …Ke7, black is giving the decision to to white. If white pushes 2.d4, black takes the pawn and plays Kd6 after 3.Kd4 to give a standard K+P vs K ending. If white pushes the king forward to e4, black again takes the direct opposition, but this time across one square from e6, and then pushing d4 is same after the exchanges. If white plays 2.Kf4, black takes the opposition from f6. That is the point of the longer distance opposition- it waits for the enemy king to come forward so that a shorter direct opposition can be taken when required.
Distant opposition as I remember.
The only hanging thread I really see is…
1. …………Ke7
2. d4 cd4
3. Kd3
If I were white, I might try this to confuse my opponent, but it would really only benefit white if black played the king to f-file, which was be very weird even for an inexperienced player to do. Any king move on the e and d-files should still draw, even the counterintuitive Kd6.