Competition problem — Combinatorics

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Combinatorics Difficulty 6.0 National Olympiad Prove it JBMO

Problem:

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

Solution

Solution:

NO. If such a configuration existed, the number of segments starting from each of the $2011$ points towards the other one and having length less than $1$ would be $1005$ . Since each segment is counted twice, their total number would be $1005 \cdot 2011 / 2$ which is not an integer, contradiction!

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.