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A position from one of my studies published in 2008:

[D] 4q3/2r1r2p/3P3P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1

White to move
(Draw! There is only one move which doesn't lose!)

The same idea, but it is very easy for engines to find the right move:

Á.Rusz (original)
[D] 6k1/1r1q4/2P3PP/8/5p1B/1p1p1P1p/1P1P3P/1K6 w - - 0 1
White to move and draw

1.dxc7/1.exd7/1.e7 Which one? Only one move draws! Very hard for engines...

[D] 3q4/2rr3p/3PP2P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1

Arpad Rusz wrote:The same idea, but it is very easy for engines to find the right move:

Á.Rusz (original)
[D] 6k1/1r1q4/2P3PP/8/5p1B/1p1p1P1p/1P1P3P/1K6 w - - 0 1
White to move and draw

Indeed very easy.
All engines find it.
The solution:

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "1-0 Easy(6)."]
[Black "?"]
[Result "1-0"]
[Annotator "George"]
[SetUp "1"]
[FEN "6k1/1r1q4/2P3PP/8/5p1B/1p1p1P1p/1P1P3P/1K6 w - - 0 1"]
[PlyCount "11"]

1. cxb7 Qxb7 2. Bf6 Qg7 3. h7+ Qxh7 4. gxh7+ Kxh7 5. Be5 Kg6 6. Bxf4 1-0

The key is to capture the Rook and not the Queen is this position:
[d]7k/6qP/5BP1/8/5p2/1p1p1P1p/1P1P3P/1K6 w - - 0 4
Where if white had captured the Queen the on g7 there would be a Rook.
So if white plays 4.Bc3 then with the Rook there it would be stalemate. Now with the Queen there it is a win for white!

Arpad Rusz wrote:A position from one of my studies published in 2008:

[D] 4q3/2r1r2p/3P3P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1

White to move
(Draw! There is only one move which doesn't lose!)

1.Rxe7 Qxe7 2.dxc7!! and draw!!!
No engine can find this.
The solution:

[Event "?"]
[Site "?"]
[Date "2010.12.23"]
[Round "?"]
[White "Draw!!!! Very tough(88)"]
[Black "?"]
[Result "1/2-1/2"]
[Annotator "George"]
[SetUp "1"]
[FEN "4q3/2r1r2p/3P3P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1"]
[PlyCount "37"]

1. Rxe7 Qxe7 (1... Rxe7 2. dxe7 Qxe7 3. Bxh7 Qe1+ 4. Bb1 $11) 2. dxc7 Qxc7 3.
Bxh7 Qc1+ 4. Bb1 Qxh6 5. Bd3 Qc1+ 6. Bb1 Qc2 7. Bxc2 bxc2 8. Ka2 c1=N+ 9. Kb1
Ne2 10. Kc2 $8 Nf4 11. Kc3 $8 Ne6 12. Kc2 $8 Nd8 13. Kc3 $8 Nc6 14. Kc2 $8 Ne7
15. Kc3 $8 Nf5 16. Kc2 $8 Ne3+ 17. Kc3 Nd5+ 18. Kc2 Nc7 19. Kc3 $8 $11 1/2-1/2

The reason that they can't find it is that after the resulting position they are still not be able to solve it. They just can't understand that black's Queens is helpless to win in this position against the Bishop. If they search 100 plies i guess they will see the draw.
[d]8/8/7q/1p6/kP6/Pp6/1P6/KB6 w - - 0 5

Arpad Rusz wrote:1.dxc7/1.exd7/1.e7 Which one? Only one move draws! Very hard for engines...

[D] 3q4/2rr3p/3PP2P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1

1.e7!! and draw!
Not very hard for engines, but impossible to understand the draw.
The solution(there is a nice little variation that black can try but white can solve it too for the draw):

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Very tough(89)"]
[Black "?"]
[Result "1/2-1/2"]
[Annotator "George"]
[SetUp "1"]
[FEN "3q4/2rr3p/3PP2P/1p6/kP6/Pp6/1P6/KB2R3 w - - 0 1"]
[PlyCount "18"]

1. e7 Qe8 2. dxc7 Rxc7 $8 (2... Rxe7 3. Rxe7 Qc8 4. Bd3 Qb7 5. c8=N (5. c8=Q
Qh1+ 6. Bb1 Qxb1+ 7. Kxb1 $11) 5... Qh1+ 6. Bb1 Qxb1+ 7. Kxb1 $11) 3. Re6 Rc6 (
3... Ra7 4. Bxh7 Qxe7 5. Rxe7 Rxe7 6. Bf5 Re1+ 7. Bb1 Rg1 8. h7 Rh1 9. h8=Q
Rxh8 10. Be4 Rc8 11. Bf5 Rc1+ 12. Bb1 $11 {Same continuation with the other
problem.}) 4. Rxc6 Qxc6 5. Bxh7 Qe8 6. Be4 Qxe7 7. Bb1 Qe1 8. h7 Qh1 9. h8=Q
Qxh8 $11 {Same continuation with the other problem.} 1/2-1/2