OlympiadBench / 1644 / Given positive integers m and n≥m, determine the largest number of…

Given positive integers m and n≥m, determine the largest number of dominoes (1×2 or 2×1 rectangles) that can be placed on a rectangular board with m rows and 2n columns consisting of cells (1×1 squares ) so that:



(i) each domino covers exactly two adjacent cells of the board;



(ii) no two dominoes overlap;



(iii) no two form a 2×2 square; and



(iv) the bottom row of the board is completely covered by n dominoes.

Given positive integers $m$ and $n \geq m$, determine the largest number of dominoes $(1 \times 2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells $(1 \times 1$ squares $)$ so that:



(i) each domino covers exactly two adjacent cells of the board;



(ii) no two dominoes overlap;



(iii) no two form a $2 \times 2$ square; and



(iv) the bottom row of the board is completely covered by $n$ dominoes.