Difference between revisions of "2020 IMO Problems/Problem 6"

(Created page with "Problem 6. Prove that there exists a positive constant c such that the following statement is true: Consider an integer n > 1, and a set S of n points in the plane such that t...")
 
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Consider an integer n > 1, and a set S of n points in the plane such that the distance between
 
Consider an integer n > 1, and a set S of n points in the plane such that the distance between
 
any two different points in S is at least 1. It follows that there is a line ℓ separating S such that the
 
any two different points in S is at least 1. It follows that there is a line ℓ separating S such that the
distance from any point of S to ℓ is at least <math>cn^−1/3</math>
+
distance from any point of S to ℓ is at least cn^(−1/3)
 
.
 
.
 
(A line ℓ separates a set of points S if some segment joining two points in S crosses ℓ.)
 
(A line ℓ separates a set of points S if some segment joining two points in S crosses ℓ.)
Note. Weaker results with <math>cn^−1/3</math>
+
Note. Weaker results with cn^(−1/3)
replaced by <math>cn^−α</math> may be awarded points depending on the value
+
replaced by cn^−α may be awarded points depending on the value
 
of the constant α > 1/3.
 
of the constant α > 1/3.

Revision as of 02:05, 23 September 2020

Problem 6. Prove that there exists a positive constant c such that the following statement is true: Consider an integer n > 1, and a set S of n points in the plane such that the distance between any two different points in S is at least 1. It follows that there is a line ℓ separating S such that the distance from any point of S to ℓ is at least cn^(−1/3) . (A line ℓ separates a set of points S if some segment joining two points in S crosses ℓ.) Note. Weaker results with cn^(−1/3) replaced by cn^−α may be awarded points depending on the value of the constant α > 1/3.