1993 OIM Problems/Problem 5

Problem

Let $P$ and $Q$ be two different points on the plane. Let us denote by $m(PQ)$ the bisector of the segment $PQ$. Let $S$ be a finite subset of the plane, with more than one element satisfying the following properties:

a. If $P$ and $Q$ are points distinct from $S$, then $m(PQ)$ intersects $S$.

b. If $P_1Q_1$, $P_2Q_2$, and $P_3Q_3$are three different segments whose ends are points of $S$, then no point of $S$ belongs simultaneously to the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$.

Determine the number of points that $S$ can have.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe8.htm