5

Part of 2022 Romania Team Selection Test

Problems(2)

All distances between points are >1 or \leq 2

Source: Romanian TST 2022, Test 1, P5

Given is an integer

k\geq 2

. Determine the smallest positive integer

n

, such that, among any

n

points in the plane, there exist

k

points among them, such that all distances between them are less than or equal to

2

, or all distances are strictly greater than

1

combinatoricsgeometry

Romania TST 2022 Day 2 P5

Source: Romania TST 2022

m,n\geq 2

be positive integers and

S\subseteq [1,m]\times [1,n]

be a set of lattice points. Prove that if

|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor

then there exists a circle which passes through at least four distinct points of

S.

combinatoricslattice pointsromaniaRomanian TST