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Problem

A reсtangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

My solution

If there exists a small rectangle with odd lengths of sides, then all distances to the sides $\mathcal{R}$ are odd. The area of such small rectangle is odd. Let's assume that there is no such small rectangle. Therefore, the areas of all small rectangles are even. But then their total area is also even. However, this is the area of $\mathcal{R}$, which means we have obtained a contradiction.

Question

The official solution seems non-trivial to me. My solution is very clear. Maybe there is an error in it. Please help me find.

Denis Krylov
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    Your first assumption is wrong. A 1x1 square in the middle of a 5x5 square has all even distances from the sides. and in the middle of the second row has even distances horizontally and odd distances vertically. – Mark Bennet Feb 22 '23 at 12:30

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