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Contests
National and Regional Contests
China Contests
China Team Selection Test
1997 China Team Selection Test
2
China TST 1997 5-element subsets
China TST 1997 5-element subsets
Source: China TST 1997, problem 5
May 22, 2005
combinatorics unsolved
combinatorics
Problem Statement
Let
n
n
n
be a natural number greater than 6.
X
X
X
is a set such that
∣
X
∣
=
n
|X| = n
∣
X
∣
=
n
.
A
1
,
A
2
,
…
,
A
m
A_1, A_2, \ldots, A_m
A
1
,
A
2
,
…
,
A
m
are distinct 5-element subsets of
X
X
X
. If
m
>
n
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
(
4
n
−
15
)
600
m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}
m
>
600
n
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
(
4
n
−
15
)
, prove that there exists
A
i
1
,
A
i
2
,
…
,
A
i
6
A_{i_1}, A_{i_2}, \ldots, A_{i_6}
A
i
1
,
A
i
2
,
…
,
A
i
6
(
1
≤
i
1
<
i
2
<
⋯
,
i
6
≤
m
)
(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)
(
1
≤
i
1
<
i
2
<
⋯
,
i
6
≤
m
)
, such that
⋃
k
=
1
6
A
i
k
=
6
\bigcup_{k = 1}^6 A_{i_k} = 6
⋃
k
=
1
6
A
i
k
=
6
.
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