Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.6 National olympiad Prove it China

An integer $n$ is called good if $n \ge 3$ and there are $n$ lattice points $P_1, P_2, \dots, P_n$ 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.

Solution

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 $P_1P_2$ 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.

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