Difference between revisions of "2023 OIM Problems/Problem 5"
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<cmath>P_{a1}P_{a2},P_{a2}P_{a3},\cdots,P_{an}P_{a1}</cmath> | <cmath>P_{a1}P_{a2},P_{a2}P_{a3},\cdots,P_{an}P_{a1}</cmath> | ||
| − | have the same length. Find the largest number <math>k</math> such that for any sequence of <math>k</math> points in the plane, | + | have the same length. Find the largest number <math>k</math> such that for any sequence of <math>k</math> points in the plane, <math>2023-k</math> points can be added so that the sequence of 2023 points is ''carioca''. |
== Solution == | == Solution == | ||
Latest revision as of 02:21, 14 December 2023
Problem
A sequence 
of points in the plane (not necessarily distinct) is "carioca" if there exists a permutation 
of the numbers 
such that all the segments
![\[P_{a1}P_{a2},P_{a2}P_{a3},\cdots,P_{an}P_{a1}\]](http://latex.artofproblemsolving.com/7/f/c/7fcab7ede7696e046a0761cb3e6c847a56e518dd.png)
have the same length. Find the largest number 
such that for any sequence of points in the plane, 
points can be added so that the sequence of 2023 points is carioca.
Solution
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See also
https://sites.google.com/associacaodaobm.org/oim-brasil-2023/pruebas
