1999 OIM Problems/Problem 6

Problem

Let $A$ and $B$ be points on the plane and $C$ be a point on the bisector of $AB$ . A sequence $C_1, C_2, \cdots , C_n, \cdots$ is constructed in the following way:

$C_1 = C$ and for $n \ge 1$ , if $C_n$ does not belong to segment $AB$ , C_{n+1} is the circumcenter of triangle $ABC_n$ .

Find all points $C$ such that the sequence $C_1, C_2, \cdots , C_n, \cdots$ is defined for all $n$ and is periodic from a certain point.

NOTE: A sequence $C_1, C_2, \cdots , C_n, \cdots$ is periodic from a certain point if there are positive integers $k$ and $p$ such that $C_{n+p} = C_n$ for all $n \ge k$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1999 OIM Problems/Problem 6

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