Problem

Source: LMAO 2024 P6

Tags: combinatorics, geometry, lmao, combinatorial geometry



There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that ___$\bullet$ Each circle in $\mathcal C$ has unit radius. ___$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. ___$\bullet$ 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