Let $P$ be a 2023 -sided polygon. All but one side has length 1 . What is the maximum possible area of $P$ ?
First, we claim $P$ must be convex to maximize its area. If not, let $A$ and $B$ be consecutive vertices on the perimeter of its convex hull that aren't
consecutive vertices of $P$. Reflecting the path between $A$ and $B$ over line $A B$ must increase the area of $P$ as the new shape strictly contains $P$.

Thus we assume $P$ is convex. Let $\ell$ be the line containing the side with length not equal to 1 . Let $P^{\prime}$ be the reflection of $P$ over $\ell$. By convexity, $P$ and $P^{\prime}$ do not overlap, so the union of $P$ and $P^{\prime}$ is a polygon, specifically an equilateral 4044-gon with sides of length 1 .

The area of an equilateral polygon is maximized when it is regular, so this union has maximum area that of a regular 4044-gon, which is $1011 \cot \pi / 4044$. Thus, the answer is half of this, i.e.

$$
\frac{1011}{2} \cdot \cot \frac{\pi}{4044}
$$