good puzzle

If Black plays 10... Ka2 11. Rb8 (11...Ka3 Ra8 mate). No matter where black moves, white will will win the knight.

I fail to see the problem with 3... Rxa3. I likewise have a problem with the fact that you must play Rb8, and not Rb7, Rb6, or Rb5.

If you mean 2...Rxa3, then 3. Rb2! Ra2 (only move), 4. Rb1#

The puzzle also seems to work with 9Rd8, f8 or h8 - not sure why the rook has to go to g8