RU-ALLRUSSIAN 1967 Problem 8

499 white rooks and a black king are placed on a $1000 \times 1000$ chess board. The rook and king moves are the same as in ordinary chess, except that taking is not allowed and the king is allowed to remain in check. No matter what the initial situation and no matter how white

moves, the black king can always:

(a) get into check (after some finite number of moves); (b) move so that apart from some initial moves, it is always in check after its move; (c) move so that apart from some initial moves, it is always in check (even just after white has moved).

Prove or disprove each of (a) - (c).