Puzzle 1 from endgame textbook

The correct move is 1. R5+2

Some general remarks: There are a couple of lines in
this position yielding an endgame rook and veteran (that's
a pawn that has reached the opponent's baseline) against
rook. This is a standard endgame, the veteran side can
only win if it's controlling the central fil. However, in
the actual position there is also a black advisor. Normally,
this additional advisor will allow that Black regains the
control of the central file or forces an exchange of the
rooks. Thus, after an exchange of the cannons Red can win
only, if he also wins Black's pawn (no problem) and the
advisor while maintaining the control of the central file.

After 1. P2.1? Black can defend the position successfully:
1. P2.1? R6+3 2. R5+2 R6.4 and Red can't win, e.g.:
A) 3.P1.2 R4-7 4.P2.3 K6+1 5.R5.3 A6-5 6.R3+4 K6+1 7.R3-4 K6-1
8.R3.4 A5+6 9.K5.4 R4+1 (Answer1)
B) 3.R5+3 K6+1 4.K5.4 R4-8 5. R5.8 (5.R5.4 K6.5 6.R4.1?? R4.6)
5. ... K6.5 (Answer2)

Red has more alternatives after 1.R5+2! and can win:
1.R5+2 R6+3 2.R5.3! (Threatening 3.P2.3 K6+1 4.R3+4 checkmate)
2. ... A6-5 3.R3+4 K6.5 (3. ... C9+8 4.R3.5 R6.7 5.P2.3 R7-8
6.R5+1 K6+1 7.R5.3, Answer4) 4.R3.5 K5.4 5.P2.1 and now Red
has reached the wanted endgame rook and veteran against rook
while controlling the central file (Answer3). This is a simple
win, e.g. 5. ... R6.8 6.R5-3 R8.4 7.P1.2 R4-2 8.P2.3 R4-1
9.P3.4 R4+1 10.P4.5 K4+1 11.R5+3 K4+1 12.P5.6 R4.3
13.R5-3 R3.4 14.P6.7 K4-1 15.R5+4 R4-1 16.R5.6

Puzzle 2 from a game Meng Wen Xuan - Wang Jie Ran played in 1917
The correct move is 1. ... R4+4

1. ... H5+4? is forking Red's king and rook, but after 2.C4.6
Black is lost:
A) 1. ... H5+4? 2.C4.6 R4+4 3.R8+5 R4-4 4.R8.6 checkmate (Answer1)
B) 1. ... H5+4? 2.C4.6 H4-2 3.C6.2 and Black has no defense
against C2+7 checkmate (Answer2)
C) 1. ... H5+4? 2.C4.6 H3+1 3.H6+5 R4+2 H5+3 K5.4 R8-4 followed by
R8+3 and R8.6 (Answer3) 
D) 1. ... H5+4? 2.C4.6 C2.4 3.H6+4 R4+3 R8+5 R4-3 H4+3 checkmate (Answer4)

Therefore, Black has to play 1. ... R4+4. After 2.R8+5 R4-4
3.R8-9 H5+4 4.K5.4 H4+6 5.K4+1 R4+3 6.R8+9 R4-3 (Answer5) a draw is
a likely result.

Puzzle 3 from a game Liu Dahua - Li Laiqun 
The correct move is 1. ... C4+7

After 1. ... C4+7 2.C1.6 2.H6+7 K5.4 3.C1.6 P4.5 Black has
to be very careful but he should be able to defend the position
successfully.

However, 1. ... C4.1? is worse, with 2.C1+7 Red will win a
piece and the game, e.g.:
A) 1. ... C4.1? 2.C1+7 C1.9 3.H4+3 checkmate (Answer1)
B) 1. ... C4.1? 2.C1+7 A5-4 3.H4+6 K5+1 4.C1.9 (Answer2)
C) 1. ... C4.1? 2.C1+7 A5+6 3.H4+6 K5.4 4.H6-5 (Answer3)
D) 1. ... C4.1? 2.C1+7 C5+2 3.C1.9 C5.4 4.K6.5 (Answer4)