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1999 OIM Problems/Problem 3

Revision as of 15:04, 13 December 2023 by Tomasdiaz (talk | contribs) (Created page with "== Problem == Let there be <math>n</math> different points, <math>P_1, P_2, \cdots , P_n</math>, on a straight line of the plane (<math>n /ge 2</math>). We consider the circle...")
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Problem

Let there be $n$ different points, $P_1, P_2, \cdots , P_n$, on a straight line of the plane ($n /ge 2$). We consider the circles of diameter $P_iP_j$ ($1 \le i < j le n$) and we color each circle with one of $k$ given colors. We call this configuration $(n, k)$-th.

For each positive integer $k$, find all $n$ for which every $(n, k)-$th is verified to contain two externally tangent circles of the same color.

NOTE: To avoid ambiguity, points that belong to more than one circle do not have a color.

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

Solution

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See also

https://www.oma.org.ar/enunciados/ibe14.htm

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