MathDB

Let

\mathcal{M}

a set of

3n\ge 3

planar points such that the maximum distance between two of these points is

1

. Prove that:
a) among any four points,there are two aparted by a distance at most

\frac{1}{\sqrt{2}} .

b) for

n=2

and any

\epsilon >0,

it is possible that

12

15

of the distances between points from

\mathcal{M}

lie in the interval

(1-\epsilon , 1];

but any

13

of the distances can´t be found all in the interval

\left(\frac{1}{\sqrt 2} ,1\right].

c) there exists a circle of diameter

\sqrt{6}

that contains

\mathcal{M} .

d) some two points of

\mathcal{M}

are on a distance not exceeding

\frac{4}{3\sqrt n-\sqrt 3} .

Problem Statement