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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2022 Moldova Team Selection Test
8
8
Part of
2022 Moldova Team Selection Test
Problems
(1)
the union of every $k$ sets contains exactly $k+1$ points
Source: Moldova TST 2022
4/1/2022
a) Let
n
n
n
(
n
≥
2
)
(n \geq 2)
(
n
≥
2
)
be an integer. On a line there are
n
n
n
distinct (pairwise distinct) sets of points, such that for every integer
k
k
k
(
1
≤
k
≤
n
)
(1 \leq k \leq n)
(
1
≤
k
≤
n
)
the union of every
k
k
k
sets contains exactly
k
+
1
k+1
k
+
1
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
k
k
k
the union of every
k
k
k
sets contains exactly
k
+
1
k+1
k
+
1
points?
combinatorics