Question 27
A rectangle $\mathcal{R}$ has all side lengths odd integers. We divide it into several smaller rectangles such that every side length of each small rectangle is a positive integer. Prove that there is at least one small rectangle whose distances to the four sides of $\mathcal{R}$ all have the same parity.
(Here, the distance from a small rectangle to a side $L$ of $\mathcal{R}$ is defined as the perpendicular distance from $L$ to the side of the small rectangle that is parallel to $L$ and closer to $L$.)
(Here, the distance from a small rectangle to a side $L$ of $\mathcal{R}$ is defined as the perpendicular distance from $L$ to the side of the small rectangle that is parallel to $L$ and closer to $L$.)