MathDB

There are two very strict laws in the country of Militaria.
(i) Anyone who is shorter than

80\%

(or more) of his “neighbours” (i.e. men living at distance less then

r

from him) is freed from the military service.
(ii) Anyone who is taller than

80\%

(or more) of his “neighbours” (i.e. men living at distance less then

R

from him) is allowed to serve in the police.
A nice thing is that each man

X

may choose his own (possibly different) positive numbers

r = r(X)

and

R = R(X)

. Can it happen that

90\%

(or more) of the men in Militaria are free from the army and, at the same time,

90\%

(or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.)
(N Konstantinov)

Problem Statement