MathDB

Given a positive integer

n \ge 2

, let

\mathcal{P}

and

\mathcal{Q}

be two sets, each consisting of

n

points in three-dimensional space. Suppose that these

2n

points are distinct. Show that it is possible to label the points of

\mathcal{P}

P_1,P_2,\dots,P_n

and the points of

\mathcal{Q}

Q_1,Q_2,\dots,Q_n

such that for any indices

i

and

j

, the balls of diameters

P_iQ_i

and

P_jQ_j

have at least one common point.

Problem Statement