Puzzle 1: Red to move and win
Solution

With his last move H2+4 Black wants to simplify the position. Red has to accept that some pieces have to leave the board, he has to look for the best transformation. At first sight it might seem that Red can only reach endgames with a material combination which looks drawish. But it's not enough to count the pieces directly after the exchange, Red can transform the position into an endgame Horse vs. 2 advisors where Black's pieces are badly coordinated. Consequently Black loses one of his advisors and the endgame with one advisor against a horse is lost.

1. P5.6      
2 points for the key move
1. . . . C4+7
  1. ... A5+4 2. H5+6 and Black's cannon is lost. 1 point
  1. ... C4.1 2. P6+1 and Red's material advantage is decisive. 1 point
2. A5+6 A5+4
 

If Black doesn't take the pawn, he will get into a hopeless endgame. Red could easily exchange his pawn against one of Black's advisors to reach the technically won endgame horse vs. advisor. On top of that Red has a couple of variations where he can win an advisor for free and force a quick checkmate, e.g. 2. ... K5.6 3.P6.5 K6+1 4.H5+3 K6-1 5.H3+2 K6+1 6.K5+1 A5-4 7.P5.4 or 2. ... K5.4 3.P6.5 K4.5 4.K5+1 K5.6 5.P5.4 A5+6 6.H5+4 K6+1 7.H4+2.

1 point, if the hopelessness of 2.... K5.6 and 2. ... K5.4 was mentioned.

3. K5+1
1 point
Now we have the endgame Horse vs. 2 advisors (Red's advisors can't help in this type of endgame). Normally the two advisors should be enough for a draw if they can protect each other. However, Black's advisors can't move at the moment and Black's king must not leave the central file. Therefore, Black's king will be forced to move to the centre of the palace, the most unfavourable place.
1 point
3. . . . K5+1
  3. . . . A6-5
  4. H5+6
  3. ... A4-5
  4. H5+4
4. K5.4
1 point
4. . . . K5.6
  4. . . . K5.4
  5. H5+4
  4. . . . K5+1
  5. H5+4
  4. . . . K5-1
  5. H5+6
5. H5+6
1 point (answer1)
Puzzle 2: Red to move and win
Solution

Most of Red's pieces are very active and Black's king is in a vulnerable position but both of Red's rooks are under attack and Red can't play 1. C3-1?? because of 1. ... R6+7 checkmate. With a sacrifice of his passive rook Red eliminates Black's dangerous horse and wins time for a decisive attack of his remaining pieces.

1. R1.6      
2 points
1. . . . R4+4
  1. ... R6.8 2.R6+4 K4.5 and Red has one horse more and is still attacking. This is clearly a won position, although a couple of variations are possible. The source, where I took this position from, doesn't even mention 1. ... R6.8, Here are two examples how the game could continue: 3.R6.5 K5.6 4.H8-6 K6-1 5.R5.4 A5+6 6.C3-2 C1.2 7.C3.4 A6-5 8.C4+3 A5+6 9.C4-2 K6-1 10.C4.3 K6.5 11.H6+4 K5.6 12.H4+3 K6.5 13.H3-2 (Answer 3) or 6. ... C1+1 7.E7+9 R8.7 8.C3.4 A6-5 9.C4+3 A5+6 10.C4-2 K6-1 11.E5+3 R7-1 12.C4-1 C1.2 13.H6+4 R7.6 14.C4+2 K6+1 15.H4+2 (Answer 4) In this problem the challenge was Red to move and win, I didn't expect a complete analysis of the position after 1. ... R6.8 2.R6+4. 2 points for everybody who made clear (with words or exemplary lines) that the position is a clear win for Red.
2. E5+7
2 points, Red's threat is 3. H8+7 checkmate
2. . . . C1.3
3. R2.4 A5+6
4. C3.9
2 points, threatening 5. C9-1 checkmate
4. . . . R4+1
5. K5+1 R4-1
6. K5-1 C3.1
7. E7+9 R4-4
Black is without defence.
8. C9-1 checkmate
1 point for end of the main line (answer1)
Puzzle 3: Red to move and checkmate
Solution

Red shouldn’t waste too much time and find a quick checkmate. As so often, Red has to clear the central file allowing Red’s king to help in the attack.

1. R3+5      
2 points
1. . . . E5-7
2. C5.6
1 point
2. . . . P3.4
  2. . . . H2+4
  3. R8+9
  1 point
  3. . . . H1-2
  4. H7+8 K4+1
  5. P6+1 mate
  1 point for the last moves of answer 2
3. C1.6 H2+4
4. R8+9
1 point
4. . . . H1-2
5. H7+8 K4+1
6. P6+1 mate
1 point for the last moves of answer 1