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Combinatorial geometry with circle interiors

Source: Romania JBMO TST 2024 Day 4 P4

July 31, 2024

combinatoricscombinatorial geometry

Problem Statement

Let

n\geqslant 2

be an integer and

A{}

a set of

n

points in the plane. Find all integers

1\leqslant k\leqslant n-1

with the following property: any two circles

C_1

and

C_2

in the plane such that

A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)

and

|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k

have at least one common point.
Cristi Săvescu

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