KöMaL 2019 K634

A sheet of graph paper has a grid of unit squares on it. A rectangle is drawn with sides lying along grid lines. Is it possible to draw a closed broken line in the rectangle along grid lines such that it should never leave the rectangle but it should pass through all the grid points in the interior and on the boundary of the rectangle, if the dimensions of the rectangle are $a)$ $2019\times2020$ units; $b)$ $2018\times2020$ units? If so, determine the length of the possible broken lines, too.