VMO 1992 Problem 4

Label the squares of a 1991 x 1992 rectangle (m, n) with 1 ≤ m ≤ 1991 and 1 ≤ n ≤ 1992. We wish to color all the squares red. The first move is to color red the squares (m, n), (m+1, n+1), (m+2, n+1) for some m < 1990, n < 1992. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?