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China Team Selection Test
2018 China Team Selection Test
6
2018 China TST 1 Day 2 Q3
2018 China TST 1 Day 2 Q3
Source: China Wuhan . Dec 31, 2017
January 2, 2018
combinatorics
inequalities
China TST
Problem Statement
Let
A
1
A_1
A
1
,
A
2
A_2
A
2
,
⋯
\cdots
⋯
,
A
m
A_m
A
m
be
m
m
m
subsets of a set of size
n
n
n
. Prove that
∑
i
=
1
m
∑
j
=
1
m
∣
A
i
∣
⋅
∣
A
i
∩
A
j
∣
≥
1
m
n
(
∑
i
=
1
m
∣
A
i
∣
)
3
.
\sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.
i
=
1
∑
m
j
=
1
∑
m
∣
A
i
∣
⋅
∣
A
i
∩
A
j
∣
≥
mn
1
(
i
=
1
∑
m
∣
A
i
∣
)
3
.
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