Here is the solution for the second puzzle.

1.fxg3 exd4

1. bxc7 Kb7
2. c6+ Kxc7
3. exd4 Kxc6
4. gxf4 Kd7
5. Kg2 Ke6
6. Kg3 Kf5
7. Kf3 d5
8. Kg3 Ke4
9. Kg4 Kxd4
10. f5 Kc5
11. f6 Kd6
12. Kg5 Ke6
13. Kg6 Kd6
14. Kg7 d4
15. f7 Kd5
16. f8=Q
    It's a win.

As I wrote, the second puzzle has a win for white, but 1.bxc7 leads only to a draw.

Here is the solution for the fourth puzzle. At last I have found a beautiful solution.

1. g8=N+ Nxg8
2. c7 Rg7+
3. Kh3 Rh7+
4. Kg2 Kd7
5. Rxf7+ Rxf7
6. e6+ Kxe6
7. c8=Q+ Kf6
   It's a draw.

Here is the solution for the second puzzle.
I can't find a win. I got only a draw.

1. bxc7 Kb7
2. fxg3 dxc5
3. dxc5 fxg3
4. Kg2 Kxc7
5. Kxg3 Kc6

In the second puzzle, can I assume you are stuck in 1.bxc7? If so, then you are studying the wrong 1st move for white.

In the forth puzzle, that line is definitely drawn, but it isn't the first move the composer of the problem had in mind either. Give the it another look. As for the 3rd puzzle, you are right- 1. ....Bd8 definitely busts the composer's idea of 1.Ka8.

Here is the solution for the third problem.
V-1

1. Ka8 b2
2. b7 Bg3
3. f4 Bxf4
4. e5 b1=Q
5. Be4+ Kxe4
6. b8=Q Qa2+
   It's a draw

V-2
1.Ka8 Bg3

1. f4 b2
2. b7 Bxf4
3. e5 Bxe5
4. b8=Q Bxb8
5. Bd5
   It's a draw

V-3

1. Ka8 Bd8
2. b7 Bc7
3. b8=Q Bxb8
4. Kxb8 b2
5. Kc7 b1=Q
   It's a loosing position for white.

I think the fourth puzzle is a draw.

1. Rg5 Rxg7
2. Kf4 Rg8
3. Rxg8 Nxg8
4. Kg5 Kd8
5. h4 Kc7
6. h5 Kxc6
7. h6 Nxh6
8. Kxh6 Kd5
9. Kg7 Kxe5
10. Kxf7

Thank you friends!

Surpriselover, sorry, but I often don't finish out the FEN string when transposing the position- just lazy. Alena, however, is correct, the missing - - 0 1 will fix all the problems you are having- just leave a space after the final b or w. I will, from now on give the entire string.

Surpriselover,

Surpriselover, you can use https://lichess.org/editor for example, or maybe put - - at the end of the FEN string

Hey Yancey, I tried to paste your first two problems on Analyze This but it says it is malformatted...could you post as diagrams or what program can I use for strings? Thanks!

Forget that comment, I thought we were talking about another problem. I think the second puzzle is a win for white, but am not 100% sure. It is the other two puzzles I am more sure are busted.

The second puzzle is a likely draw, and thus a busted puzzle.

Here is the solution for the first puzzle.
It was easy for me.I managed to get a draw.
V-1

1. Rg8+ Qe8
2. Rh8 Qxh8

V-2

1. Rg8+ Qe8
2. Rh8 b5
3. Rxe8+ Kxe8
4. Kc5 Ke7
5. Kxb5 Kf7
6. d6 Ke6
7. Kc4 Kxd6
8. Kd4 (we got the opposition)
   It's more complicated. You need to know pawn endings well.

I have tried to solve the second puzzle but I only got a draw in different ways.

Puzzles for Alena:
3k4/3p2R1/1p1K4/3P4/8/8/8/4q3 w How should white proceed?

k7/2p5/1P1p4/2P1p3/3P1p2/4P1p1/5P2/7K w What do you make of this problem that is described as "White to move and win"?

8/1K6/1P6/8/4P2b/1p1k1P2/8/7B w What do you make of this problem that is described at "White to move and draw"?

8/4kpPr/2P4n/4PR2/8/6K1/7P/8 w What do you make of this problem that is described as "White to move and win"?

1. Qf8+ Kxf8
2. Rd8#

or

1. Qf8+ Kh7
2. Qg7#