MathDB

1. Consider in the plane a set

H

of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers

n

, the circle

k_n

with centre at the origin and of radius

n

contains at least

cn^2

elements of the set

H

. Prove that there exists a straight line which intersects infinitely many of the circles of

H

. Show further that if we require only that the circles

k_n

contain o(n²) elements of

H

, the proposition will be false. (G. 5)

Problem Statement