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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2010 China Second Round Olympiad
2010 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
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2010 China Second Round,test 2,problem 4
the code system of a new 'MO lock' is a regular
n
n
n
-gon,each vertex labelled a number
0
0
0
or
1
1
1
and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide. find the number of all possible codes(in terms of
n
n
n
).
3
1
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2010 China Second Round,test 2,problem 3
let
n
>
2
n>2
n
>
2
be a fixed integer.positive reals
a
i
≤
1
a_i\le 1
a
i
≤
1
(for all
1
≤
i
≤
n
1\le i\le n
1
≤
i
≤
n
).for all
k
=
1
,
2
,
.
.
.
,
n
k=1,2,...,n
k
=
1
,
2
,
...
,
n
,let
A
k
=
∑
i
=
1
k
a
i
k
A_k=\frac{\sum_{i=1}^{k}a_i}{k}
A
k
=
k
∑
i
=
1
k
a
i
prove that
∣
∑
k
=
1
n
a
k
−
∑
k
=
1
n
A
k
∣
<
n
−
1
2
|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}
∣
∑
k
=
1
n
a
k
−
∑
k
=
1
n
A
k
∣
<
2
n
−
1
.
2
1
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2010 China Second Round,test 2,problem 2
Given a fixed integer
k
>
0
,
r
=
k
+
0.5
k>0,r=k+0.5
k
>
0
,
r
=
k
+
0.5
,define
f
1
(
r
)
=
f
(
r
)
=
r
[
r
]
,
f
l
(
r
)
=
f
(
f
l
−
1
(
r
)
)
(
l
>
1
)
f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)
f
1
(
r
)
=
f
(
r
)
=
r
[
r
]
,
f
l
(
r
)
=
f
(
f
l
−
1
(
r
))
(
l
>
1
)
where
[
x
]
[x]
[
x
]
denotes the smallest integer not less than
x
x
x
. prove that there exists integer
m
m
m
such that
f
m
(
r
)
f^m(r)
f
m
(
r
)
is an integer.
1
1
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2010 China Second Round,test 2,problem 1
Given an acute triangle whose circumcenter is
O
O
O
.let
K
K
K
be a point on
B
C
BC
BC
,different from its midpoint.
D
D
D
is on the extension of segment
A
K
,
B
D
AK,BD
A
K
,
B
D
and
A
C
AC
A
C
,
C
D
CD
C
D
and
A
B
AB
A
B
intersect at
N
,
M
N,M
N
,
M
respectively.prove that
A
,
B
,
D
,
C
A,B,D,C
A
,
B
,
D
,
C
are concyclic.