CGMO 2005 Problem 6

An integer $n$ is called good if there are $n \geq 3$ lattice points $P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $P_iP_j$ has a rational length, then there is $P_k$ such that both line segments $P_iP_k$ and $P_jP_k$ have irrational lengths; and if line segment $P_iP_j$ has an irrational length, then there is $P_k$ such that both line segments $P_iP_k$ and $P_jP_k$ have rational lengths.

(1) Determine the minimum good number.

(2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)