Difference between revisions of "1989 IMO Problems/Problem 5"

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==Problem==
 
==Problem==
Let <math>n\geq3</math> and consider a set <math>E</math> of <math>2n − 1</math> distinct points on a circle. Suppose
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Let <math>n\geq3</math> and consider a set <math>E</math> of <math>2n − 1</math> distinct points on a circle. Suppose that exactly <math>k</math> of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly <math>n</math> points from set <math>E</math>. Find the smallest value of <math>k</math> such that every such coloring of <math>k</math> points of <math>E</math> is good.
that exactly <math>k</math> of these points are to be colored black. Such a coloring is “good”
 
if there is at least one pair of black points such that the interior of one of the
 
arcs between them contains exactly <math>n</math> points from set <math>E</math>. Find the smallest value
 
of <math>k</math> such that every such coloring of <math>k</math> points of <math>E</math> is good.
 

Revision as of 11:51, 17 June 2020

Problem

Let $n\geq3$ and consider a set $E$ of $2n − 1$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:− not set up for use with LaTeX.) distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from set $E$. Find the smallest value of $k$ such that every such coloring of $k$ points of $E$ is good.

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