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Contests
National and Regional Contests
China Contests
China Team Selection Test
2009 China Team Selection Test
2009 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(6)
4
1
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Square of rational number
Let positive real numbers
a
,
b
a,b
a
,
b
satisfy b \minus{} a > 2. Prove that for any two distinct integers
m
,
n
m,n
m
,
n
belonging to
[
a
,
b
)
,
[a,b),
[
a
,
b
)
,
there always exists non-empty set
S
S
S
consisting of certain integers belonging to [ab,(a \plus{} 1)(b \plus{} 1)) such that
∏
x
∈
S
m
n
\frac {\displaystyle\prod_{x\in S}}{mn}
mn
x
∈
S
∏
is square of a rational number.
5
1
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Minimal value
Let
m
>
1
m > 1
m
>
1
be an integer,
n
n
n
is an odd number satisfying
3
≤
n
<
2
m
,
3\le n < 2m,
3
≤
n
<
2
m
,
number
a
i
,
j
(
i
,
j
∈
N
,
1
≤
i
≤
m
,
1
≤
j
≤
n
)
a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)
a
i
,
j
(
i
,
j
∈
N
,
1
≤
i
≤
m
,
1
≤
j
≤
n
)
satisfies
(
1
)
(1)
(
1
)
for any
1
≤
j
≤
n
,
a
1
,
j
,
a
2
,
j
,
⋯
,
a
m
,
j
1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}
1
≤
j
≤
n
,
a
1
,
j
,
a
2
,
j
,
⋯
,
a
m
,
j
is a permutation of
1
,
2
,
3
,
⋯
,
m
;
(
2
)
1,2,3,\cdots,m; (2)
1
,
2
,
3
,
⋯
,
m
;
(
2
)
for any 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1 holds. Find the minimal value of
M
M
M
, where M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.
6
1
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Arithmetic progression
Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form 2^m \plus{} 3^n or not. where
m
,
n
m,n
m
,
n
are nonnegative integers.
1
7
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2
7
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3
7
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