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Contests
National and Regional Contests
China Contests
China Team Selection Test
2008 China Team Selection Test
2008 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(6)
5
1
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Chinese TST 2008 P5
For two given positive integers
m
,
n
>
1
m,n > 1
m
,
n
>
1
, let
a
i
j
(
i
=
1
,
2
,
⋯
,
n
,
j
=
1
,
2
,
⋯
,
m
)
a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)
a
ij
(
i
=
1
,
2
,
⋯
,
n
,
j
=
1
,
2
,
⋯
,
m
)
be nonnegative real numbers, not all zero, find the maximum and the minimum values of
f
f
f
, where
f
=
n
∑
i
=
1
n
(
∑
j
=
1
m
a
i
j
)
2
+
m
∑
j
=
1
m
(
∑
i
=
1
n
a
i
j
)
2
(
∑
i
=
1
n
∑
j
=
1
m
a
i
j
)
2
+
m
n
∑
i
=
1
n
∑
j
=
1
m
a
i
j
2
.
f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}.
f
=
(
∑
i
=
1
n
∑
j
=
1
m
a
ij
)
2
+
mn
∑
i
=
1
n
∑
j
=
1
m
a
ij
2
n
∑
i
=
1
n
(
∑
j
=
1
m
a
ij
)
2
+
m
∑
j
=
1
m
(
∑
i
=
1
n
a
ij
)
2
.
4
1
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Chinese TST 2008 P4
Prove that for arbitary positive integer
n
≥
4
n\geq 4
n
≥
4
, there exists a permutation of the subsets that contain at least two elements of the set G_{n} \equal{} \{1,2,3,\cdots,n\}: P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1} such that |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.
2
7
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6
1
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Chinese TST 2008 P6
Find the maximal constant
M
M
M
, such that for arbitrary integer
n
≥
3
,
n\geq 3,
n
≥
3
,
there exist two sequences of positive real number
a
1
,
a
2
,
⋯
,
a
n
,
a_{1},a_{2},\cdots,a_{n},
a
1
,
a
2
,
⋯
,
a
n
,
and
b
1
,
b
2
,
⋯
,
b
n
,
b_{1},b_{2},\cdots,b_{n},
b
1
,
b
2
,
⋯
,
b
n
,
satisfying (1): \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1; (2): a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M.
3
7
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1
7
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