Difference between revisions of "2000 OIM Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | A regular polygon with <math>n</math> sides (<math>n \ge 3</math>) is constructed and its vertices are numbered from 1 to <math>n</math>. All diagonals of the polygon are drawn. Show that if n is odd, each side and each diagonal can be assigned an integer from 1 to n, such that the following conditions are simultaneously met: | + | A regular polygon with <math>n</math> sides (<math>n \ge 3</math>) is constructed and its vertices are numbered from 1 to <math>n</math>. All diagonals of the polygon are drawn. Show that if <math>n</math> is odd, each side and each diagonal can be assigned an integer from 1 to <math>n</math>, such that the following conditions are simultaneously met: |
1. The number assigned to each side or diagonal is different from those assigned to the vertices it joins. | 1. The number assigned to each side or diagonal is different from those assigned to the vertices it joins. | ||
Latest revision as of 15:12, 13 December 2023
Problem
A regular polygon with 
sides (
) is constructed and its vertices are numbered from 1 to . All diagonals of the polygon are drawn. Show that if is odd, each side and each diagonal can be assigned an integer from 1 to , such that the following conditions are simultaneously met:
1. The number assigned to each side or diagonal is different from those assigned to the vertices it joins.
2. For each vertex, all sides and diagonals that share that vertex have different numbers.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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