Difference between revisions of "1999 OIM Problems/Problem 3"
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== Problem == | == 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 circles of diameter <math>P_iP_j</math> (<math>1 \le i < j le n</math>) and we color each circle with one of <math>k</math> given colors. We call this configuration <math>(n, k)</math>-th. | + | 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 circles of diameter <math>P_iP_j</math> (<math>1 \le i < j \le n</math>) and we color each circle with one of <math>k</math> given colors. We call this configuration <math>(n, k)</math>-th. |
For each positive integer <math>k</math>, find all <math>n</math> for which every <math>(n, k)-</math>th is verified to contain two externally tangent circles of the same color. | For each positive integer <math>k</math>, find all <math>n</math> for which every <math>(n, k)-</math>th is verified to contain two externally tangent circles of the same color. | ||
Revision as of 15:04, 13 December 2023
Problem
Let there be 
different points, 
, on a straight line of the plane (
). We consider the circles of diameter 
(
) and we color each circle with one of 
given colors. We call this configuration 
-th.
For each positive integer , find all for which every 
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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