Problem 1657

1. Define Rational and Irrational Points:
- A point $(x, y)$ in the Cartesian plane is called a *rational point* if both $x$ and $y$ are rational numbers.
- Otherwise, it is called an *irrational point*.

2. Work in the Complex Plane:
- Consider the Cartesian plane as the complex plane $\mathbb{C}$.
- Rational points correspond to points in $\mathbb{Q}[i]$, where $\mathbb{Q}[i]$ is the set of complex numbers with rational real and imaginary parts.

3. Lemma 1: Circumcenter of Rational Points:
- If $x, y, z$ are three distinct non-collinear points in $\mathbb{Q}[i]$, then the circumcenter of $\triangle xyz$ is in $\mathbb{Q}[i]$.
- Proof of Lemma 1:
- The circumcenter $t$ of $\triangle xyz$ satisfies two linear equations in $t$ and $\overline{t}$ with coefficients in $\mathbb{Q}[i]$.
- Solving these equations, $t$ must be in $\mathbb{Q}[i]$.

4. **Lemma 2: Primitive Roots of Unity in $\mathbb{Q}[i]$:**
- If $\zeta \in \mathbb{Q}[i]$ is a primitive $n$-th root of unity, then $n = 1, 2, \text{ or } 4$.
- Proof of Lemma 2:
- Primitive $n$-th roots of unity are of the form $e^{2\pi i k/n}$.
- For $\zeta$ to be in $\mathbb{Q}[i]$, both $\cos(2\pi k/n)$ and $\sin(2\pi k/n)$ must be rational.
- This only happens for $n = 1, 2, 4$.

5. **Regular $n$-gon with Rational Vertices:**
- Consider a regular $n$-gon $\{z_1, z_2, \ldots, z_n\}$ with vertices in $\mathbb{C}$.
- Suppose there exist three distinct indices $i, j, k$ such that $z_i, z_j, z_k \in \mathbb{Q}[i]$.
- By Lemma 1, the circumcenter $z$ of the polygon must be in $\mathbb{Q}[i]$.

6. Mapping to Roots of Unity:
- Consider the map $x \mapsto \frac{x - z}{z_i - z}$, which preserves $\mathbb{Q}[i]$ and the regularity of the polygon.
- This maps $\{z_1, z_2, \ldots, z_n\}$ to a permutation of $\{1, \zeta, \zeta^2, \ldots, \zeta^{n-1}\}$, where $\zeta$ is a primitive $n$-th root of unity.

7. Rationality of Roots of Unity:
- If $\{z_j, z_k\} = \{\zeta^l, \zeta^m\}$ for some $l, m \in \{1, 2, \ldots, n-1\}$, then $\zeta^l, \zeta^m \in \mathbb{Q}[i]$.
- By Lemma 2, $\frac{n}{\gcd(n, l)}$ and $\frac{n}{\gcd(n, m)}$ must be $1, 2, \text{ or } 4$.

8. **Contradiction for $n = 5$:**
- For a regular pentagon ($n = 5$), $4 \nmid 5$, leading to a contradiction.
- Therefore, there cannot be more than two rational points among the vertices of a regular pentagon.