2023 OIM Problems/Problem 5

Problem

A sequence $P_1, \cdots , P_n$ of points in the plane (not necessarily distinct) is "carioca" if there exists a permutation $a_1, \cdots , a_n$ of the numbers $1, \cdots, n$ such that all the segments

$P_{a1}P_{a2},P_{a2}P_{a3},\cdots,P_{an}P_{a1}$

have the same length. Find the largest number $k$ such that for any sequence of $k$ points in the plane, $2023 − k$ (Error compiling LaTeX. Unknown error_msg) points can be added so that the sequence of 2023 points is carioca.

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2023 OIM Problems/Problem 5

Problem

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