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 $YZ = 3XZ.$
Problem
Source: IMO LongList 1967, Socialists Republic Of Czechoslovakia 6
Tags: geometry, point set, Sequence, IMO Shortlist, IMO Longlist