1. Parametrization of the Unit Circle:
Consider the unit circle centered at the origin with radius 1. A rational parametrization of the unit circle is given by:
$$x = \frac{2t}{1 + t^2}, \quad y = \frac{1 - t^2}{1 + t^2}$$
where $t$ is a parameter. If $t$ is rational, then both $x$ and $y$ are rational.

2. Choosing Rational Points:
Choose $n$ distinct rational values for $t$, say $t_1, t_2, \ldots, t_n$. This gives us $n$ distinct rational points on the unit circle:
$$M_k = \left( \frac{2t_k}{1 + t_k^2}, \frac{1 - t_k^2}{1 + t_k^2} \right) \quad \text{for} \quad k = 1, 2, \ldots, n$$

3. Finding the Least Common Multiple (LCM):
Let $m$ be the least common multiple (LCM) of the denominators of the coordinates $x_k$ and $y_k$ for $k = 1, 2, \ldots, n$. This ensures that when we scale the coordinates by $m$, we obtain integer coordinates.

4. Scaling the Points:
Apply a homothety (scaling transformation) with center $O$ (the origin) and ratio $m$. The coordinates of the points $M_k$ after scaling by $m$ become:
$$M_k' = \left( m \cdot \frac{2t_k}{1 + t_k^2}, m \cdot \frac{1 - t_k^2}{1 + t_k^2} \right)$$
Since $m$ is the LCM of the denominators, the coordinates of $M_k'$ are integers.

5. Conclusion:
The scaled points $M_k'$ are distinct integer points on a circle centered at the origin with radius $r = m$. Therefore, we have found a radius $r$ such that the circle passes through at least $n$ distinct integer points.

$$\blacksquare$$