Difference between revisions of "2004 OIM Problems/Problem 6"
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| − | For a set <math>H</math> of points in the plane, a point <math>P</math> in the plane is said to be a point | + | For a set <math>H</math> of points in the plane, a point <math>P</math> in the plane is said to be a "''cut point''" of <math>H</math> if there are four different points <math>A, B, C</math> and <math>D</math> in <math>H</math> such that the lines <math>AB</math> and <math>CD</math> are different and intersect at <math>P</math>. Given a finite set <math>A_0</math> of points in the plane, a sequence of sets is constructed <math>A_1, A_2, A_3, \cdots</math> as follows: for any <math>j \ge 0</math>, <math>A_{j+1}</math> is the union of <math>A_j</math> with the set of all ''cut points'' of <math>A_j</math>. Show that if the union of all the sets of the sequence is a finite set, then for any <math>j \ge 1</math> we have <math>A_j = A_1</math>. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 17:36, 14 December 2023
Problem
For a set 
of points in the plane, a point 
in the plane is said to be a "cut point" of if there are four different points 
and 
in such that the lines 
and 
are different and intersect at . Given a finite set 
of points in the plane, a sequence of sets is constructed 
as follows: for any 
, 
is the union of 
with the set of all cut points of . Show that if the union of all the sets of the sequence is a finite set, then for any 
we have 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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