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Miklós Schweitzer
1960 Miklós Schweitzer
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1
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1960 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1960- Problem 1
Source:
11/18/2015
1. Consider in the plane a set
H
H
H
of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers
n
n
n
, the circle
k
n
k_n
k
n
with centre at the origin and of radius
n
n
n
contains at least
c
n
2
cn^2
c
n
2
elements of the set
H
H
H
. Prove that there exists a straight line which intersects infinitely many of the circles of
H
H
H
. Show further that if we require only that the circles
k
n
k_n
k
n
contain o(n²) elements of
H
H
H
, the proposition will be false. (G. 5)
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