MathDB

There are

n > 1

distinct points marked in the plane. Prove that there exists a set of circles

\mathcal C

such that
[color=#FFFFFF]___

\bullet

Each circle in

\mathcal C

has unit radius. [color=#FFFFFF]___

\bullet

Every marked point lies in the (strict) interior of some circle in

\mathcal C

. [color=#FFFFFF]___

\bullet

There are less than

0.3n

pairs of circles in

\mathcal C

that intersect in exactly

2

points.
Note: Weaker results with

\it{0.3n}

replaced by

\it{cn}

may be awarded points depending on the value of the constant

\it{c > 0.3}

.
Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param

Problem Statement