Problem 1673

Solution:

We subdivide $\mathcal{R}$ into unit squares and color some of these unit squares red or blue, according to the following illustration.

Since $\mathcal{R}$ has odd side lengths by assumption, all four corner squares of $\mathcal{R}$ are colored blue, and there are in total more colored than uncolored squares. Thus at least one of the rectangles $\mathcal{R}_{1}, \ldots, \mathcal{R}_{k}$ into which $\mathcal{R}$ was divided also contains more colored than uncolored squares; let $\mathcal{R}_{i}$ be such a rectangle. Then $\mathcal{R}_{i}$ has odd side lengths and all four corner squares of $\mathcal{R}_{i}$ are colored. It follows that all four corner squares of $\mathcal{R}_{i}$ must carry the same color. If they are blue, then $\mathcal{R}_{i}$ has even distances to all four sides of $\mathcal{R}$; if they are red, then $\mathcal{R}_{i}$ has odd distances to all four sides of $\mathcal{R}$. In either case, $\mathcal{R}_{i}$ satisfies the required condition.