MathDB

Let

n

be a positive integer and let

x_1,\ldots,x_n

n

nonzero points in

\mathbb R^2

. Suppose

\langle x_i,x_j\rangle

(scalar or dot product) is a rational number for all

i,j

(

1\le i,j\le n

). Let

S

denote all points of

\mathbb R^2

of the form

\sum_{i=1}^na_ix_i

where the

a_i

are integers. A closed disk of radius

R

and center

P

is the set of points at distance at most

R

from

P

(includes the points distance

R

from

P

). Prove that there exists a positive number

R

and closed disks

D_1,D_2,\ldots

of radius

R

such that
(a) Each disk contains exactly two points of

S

; (b) Every point of

S

lies in at least one disk; (c) Two distinct disks intersect in at most one point.

Problem 7