Given a segment of the length 1, define the set of points in the following way: it contains two points and also all points obtained from by iterating the following rule: With every pair of points the set contains also the point of the segment for which
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Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
Izvor: Međunarodna matematička olimpijada, shortlist 1967
Školjka 
of the length 1, define the set 
of points in the 
and also all points obtained from 
by iterating the following rule: With every pair of points 
the set 
of the segment 
for which 