I was able to solve this chess puzzle without too much difficulty.
First I looked at having the King run, and there are no safe squares where he can shield from rook checks. Then I explored the idea of giving up some or all of the pawns. I realized that giving up some pawns allows the King to hide temporarily, zig-zagging around the pawns, but this doesn't solve the problem. So I got thinking that the rook on H3 must be in the puzzle for some reason, and if we can liberate it then it could be the mating piece on a3.