Problem 1321

To determine whether there exists a convex $2023$-gon with vertices at integer points and all sides of equal length, we need to analyze the possible values of the squared side length $s$.

1. **Checkerboard Coloring Argument for Odd $s$:**
- Suppose the common side length is $\sqrt{s}$, where $s$ is odd.
- Color the lattice points in a checkerboard pattern, where each point is either black or white, and adjacent points have different colors.
- In a convex $2023$-gon, each vertex must be connected to two other vertices, forming sides of the polygon.
- Since $s$ is odd, the distance between any two adjacent vertices (which is $\sqrt{s}$) must connect points of different colors.
- However, since $2023$ is odd, there would be an odd number of sides, implying an odd number of color changes. This is impossible because the polygon must return to its starting point, requiring an even number of color changes.
- Therefore, no such polygon exists if $s$ is odd.

2. **Row Coloring Argument for $s \equiv 2 \pmod{4}$:**
- Suppose $s \equiv 2 \pmod{4}$.
- Color the rows of the lattice alternately black and white.
- For any side of length $\sqrt{s}$, the squared distance $s = \Delta x^2 + \Delta y^2$ must hold.
- Given $s \equiv 2 \pmod{4}$, both $\Delta x$ and $\Delta y$ must be odd (since the sum of two squares is $2 \pmod{4}$ only if both squares are $1 \pmod{4}$).
- This implies that any side of the polygon connects points in different rows, hence different colors.
- Again, since $2023$ is odd, there would be an odd number of color changes, which is impossible for a closed polygon.
- Therefore, no such polygon exists if $s \equiv 2 \pmod{4}$.

3. **Scaling Argument for $s \equiv 0 \pmod{4}$:**
- Suppose $s \equiv 0 \pmod{4}$.
- Then both $\Delta x$ and $\Delta y$ must be even (since the sum of two squares is $0 \pmod{4}$ only if both squares are $0 \pmod{4}$).
- This means we can write $\Delta x = 2a$ and $\Delta y = 2b$ for some integers $a$ and $b$.
- The side length $\sqrt{s}$ can be scaled down by a factor of $2$, resulting in a new polygon with side length $\sqrt{\frac{s}{4}}$.
- Repeating this process, we eventually reduce the problem to one of the previous cases (either $s$ becomes odd or $s \equiv 2 \pmod{4}$).
- Since we have already shown that no such polygon exists in those cases, it follows that no such polygon exists if $s \equiv 0 \pmod{4}$ either.

Since we have exhausted all possible cases for $s$, we conclude that no convex $2023$-gon with vertices at integer points and all sides of equal length exists.