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International Contests
APMO
1998 APMO
1
Elements of set
Elements of set
Source: APMO 1998
March 17, 2006
combinatorics unsolved
combinatorics
Problem Statement
Let
F
F
F
be the set of all
n
n
n
-tuples
(
A
1
,
…
,
A
n
)
(A_1, \ldots, A_n)
(
A
1
,
…
,
A
n
)
such that each
A
i
A_{i}
A
i
is a subset of
{
1
,
2
,
…
,
1998
}
\{1, 2, \ldots, 1998\}
{
1
,
2
,
…
,
1998
}
. Let
∣
A
∣
|A|
∣
A
∣
denote the number of elements of the set
A
A
A
. Find
∑
(
A
1
,
…
,
A
n
)
∈
F
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣
\sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n|
(
A
1
,
…
,
A
n
)
∈
F
∑
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣
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