Kenilworth Kibitzer

For the first puzzle (rook mate), it's an exercise in opposition and outflanking.  The key is that White has to maintain the opposition on the b-file and outflank on the c-file (temporarily giving up the opposition).  This is often the case in these positions - the opposition is useful only insofar as it can be converted to something more tangible, in this case better king position.  Black can only choose whether to be mated on the a-file or 8th rank...

1. Ka2! Kb8 2. Kb2 (maintaining opposition) Ka8 3. Kc3! Kb7 4. Kb3 Ka7 5. Kc4 Kb8 6. Kb4 Ka8 7. Kc5 Kb78. Kb5 Ka7 9. Kc6 Kb8 10. Kb6 Ka8 11. Rc8#

If Black sits on the a-file instead then White calmly advances up the c-file and mates on the a-file.

Second position:  White must maneuver to win the c-pawn but first keep Black's king as far from his a-pawn as possible.  Meanwhile, the d4 square is untouchable for White in the key line, what Dvoretsky calls a 'mined square'.  Let's look....

1...Kb7/Ka7 2. Kb3! Ka6 3. Kb4! (triangulation) Kb7 4. Kc4 Ka6 5. Kd3!! (5. Kd4? Ke6 is zugzwang) Ka5 6. Ke4 Kb5 7. Kd4 Ka4 8. Ke5 Kxa3 9. Kd6 and it's curtains.