A finite set of unit circles is given in a plane such that the area of their union is . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that
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A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$
Izvor: Međunarodna matematička olimpijada, shortlist 1981
Školjka 
is 
. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that 