Question 172
Let $n$ be a positive integer and consider a square of dimensions $2^{n} \times 2^{n}$. We cover this square with a number (at least 2) of non-overlapping rectangles, such that every rectangle has integer dimensions and an area that is a power of two. Prove that two of the rectangles in the covering have the same dimensions. (Two rectangles have the same dimensions if they have the same width and the same height, where they may not be rotated.)