1. Let's consider the dissections of a square with side 2012 into rectangles where one side is of length 1, and the other side is of integer length. What is the maximum number of such dissections that can be made so that no two dissections have identical rectangles in the same position?
1. For the rectangle that covers the lower left corner, there are 4023 possibilities. On the other hand, an example of 4023 partitions that satisfy the conditions is shown in the figure. Therefore, the answer is 4023.
![](https://cdn.mathpix.com/cropped/2024_06_04_2726a9c5153f9d3c9cfcg-2.jpg?height=406&width=1444&top_left_y=682&top_left_x=318)