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Difference between revisions of "2017 JBMO Problems/Problem 4"

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== Problem ==
 
== Problem ==
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Consider a regular 2n-gon <math> P</math>,<math>A_1,A_2,\cdots ,A_{2n}</math> in the plane ,where <math>n</math> is a positive integer . We say that a point <math>S</math> on one of the sides of  <math>P</math> can be seen from a point <math>E</math> that is external to <math>P</math> , if the line segment <math>SE</math> contains no other points that lie  on the sides of <math>P</math> except  <math>S</math> .We color the sides of <math>P</math> in 3 different colors (ignore the vertices of <math>P</math>,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover  ,from every point in the plane external to <math>P</math> , points of most 2  different colors on <math>P</math> can be seen .Find the number of distinct such colorings of <math>P</math> (two colorings are considered distinct if at least one of sides is colored differently).
  
 
== Solution ==
 
== Solution ==
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{{JBMO box|year=2017|num-b=3|after=Last Problem|five=}}
 
{{JBMO box|year=2017|num-b=3|after=Last Problem|five=}}
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[[Category:Intermediate Combinatorics Problems]]

Latest revision as of 14:41, 17 September 2017

Problem

Consider a regular 2n-gon $P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).

Solution

See also

2017 JBMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Last Problem
1 • 2 • 3 • 4
All JBMO Problems and Solutions
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