the union of every $k$ sets contains exactly $k+1$ points
Source: Moldova TST 2022
April 1, 2022
combinatorics
Problem Statement
a) Let be an integer. On a line there are distinct (pairwise distinct) sets of points, such that for every integer the union of every sets contains exactly points. Show that there is always a point that belongs to every set.
b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer the union of every sets contains exactly points?