China Girls' Mathematical Olympiad

We claim that the minimum good number is $5$, and that $2005$ is good.

It is not difficult to see that $n=3$ is not a good number. Note that $n=4$ is also not a good number. Assume on the contrary that there are lattice points $P_1,P_2,P_3,P_4$ satisfying the conditions of the problem. Without loss of generality, we assume that $P_1{P}_2$ is rational and $(P_1,P_2,P_3)$ is good. Then $(P_2,P_3,P_4)$ must be good. Neither $(P_2,P_4,P_1)$ nor $(P_2,P_4,P_3)$ is good, violating the condition of the problem. Hence our assumption was wrong and $n=4$ is not a good number.

For $n=5$, we define the set $$S_5=\{P_1=(0,0),P_2=(1,0),P_3=(5,3),P_4=(8,7),P_5=(0,7)\}.$$

It is not difficult to see that all triples $(P_i,P_j,P_k)$ are good. Hence $n=5$ is minimum good number.