Difference between revisions of "1993 OIM Problems/Problem 5"
(Created page with "== Problem == Let <math>P</math> and <math>Q</math> be two different points on the plane. Let us denote by <math>m(PQ)</math> the bisector of the segment <math>PQ</math>. Let...") |
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a. If <math>P</math> and <math>Q</math> are points distinct from <math>S</math>, then <math>m(PQ)</math> intersects <math>S</math>. | a. If <math>P</math> and <math>Q</math> are points distinct from <math>S</math>, then <math>m(PQ)</math> intersects <math>S</math>. | ||
| − | b. If <math>P_1Q_1</math>, <math>P_2Q_2</math>and <math>P_3Q_3</math>are three different segments whose ends are points of <math>S</math>, then no point of <math>S</math> belongs simultaneously to the three lines <math>m(P_1Q_1)</math>, <math>m(P_2Q_2)</math> | + | b. If <math>P_1Q_1</math>, <math>P_2Q_2</math>, and <math>P_3Q_3</math>are three different segments whose ends are points of <math>S</math>, then no point of <math>S</math> belongs simultaneously to the three lines <math>m(P_1Q_1)</math>, <math>m(P_2Q_2)</math>, and <math>m(P_3Q_3)</math>. |
Determine the number of points that <math>S</math> can have. | Determine the number of points that <math>S</math> can have. | ||
Latest revision as of 13:19, 13 December 2023
Problem
Let 
and 
be two different points on the plane. Let us denote by 
the bisector of the segment 
. Let 
be a finite subset of the plane, with more than one element satisfying the following properties:
a. If and are points distinct from , then intersects .
b. If 
, 
, and 
are three different segments whose ends are points of , then no point of belongs simultaneously to the three lines 
, 
, and 
.
Determine the number of points that can have.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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