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...") |
|||
| Line 2: | Line 2: | ||
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 | + | 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 | + | Note. Weaker results with cn^(−1/3) |
| − | replaced by | + | 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 01:05, 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.
