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Romania Team Selection Test
2013 Romania Team Selection Test
2013 Romania Team Selection Test
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Romania Team Selection Test
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Infinite number of sets with an intersection property
Let
k
k
k
be a positive integer larger than
1
1
1
. Build an infinite set
A
\mathcal{A}
A
of subsets of
N
\mathbb{N}
N
having the following properties:(a) any
k
k
k
distinct sets of
A
\mathcal{A}
A
have exactly one common element; (b) any
k
+
1
k+1
k
+
1
distinct sets of
A
\mathcal{A}
A
have void intersection.
Partition a (4n-1)-gon
Let
n
n
n
be an integer greater than 1. The set
S
S
S
of all diagonals of a
(
4
n
−
1
)
\left( 4n-1\right)
(
4
n
−
1
)
-gon is partitioned into
k
k
k
sets,
S
1
,
S
2
,
…
,
S
k
,
S_{1},S_{2},\ldots ,S_{k},
S
1
,
S
2
,
…
,
S
k
,
so that, for every pair of distinct indices
i
i
i
and
j
,
j,
j
,
some diagonal in
S
i
S_{i}
S
i
crosses some diagonal in
S
j
;
S_{j};
S
j
;
that is, the two diagonals share an interior point. Determine the largest possible value of
k
k
k
in terms of
n
.
n.
n
.
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