MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2009 China National Olympiad
2009 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
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The number of convex m polygon
Given two integers
m
,
n
m,n
m
,
n
satisfying
4
<
m
<
n
.
4 < m < n.
4
<
m
<
n
.
Let A_{1}A_{2}\cdots A_{2n \plus{} 1} be a regular 2n\plus{}1 polygon. Denote by
P
P
P
the set of its vertices. Find the number of convex
m
m
m
polygon whose vertices belongs to
P
P
P
and exactly has two acute angles.
Set again
Given an integer
n
>
3.
n > 3.
n
>
3.
Prove that there exists a set
S
S
S
consisting of
n
n
n
pairwisely distinct positive integers such that for any two different non-empty subset of
S
S
S
:
A
,
B
,
∑
x
∈
A
x
∣
A
∣
A,B, \frac {\sum_{x\in A}x}{|A|}
A
,
B
,
∣
A
∣
∑
x
∈
A
x
and
∑
x
∈
B
x
∣
B
∣
\frac {\sum_{x\in B}x}{|B|}
∣
B
∣
∑
x
∈
B
x
are two composites which share no common divisors.
2
2
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Prime number
Find all the pairs of prime numbers
(
p
,
q
)
(p,q)
(
p
,
q
)
such that pq|5^p\plus{}5^q.
Coloring
Let
P
P
P
be a convex
n
n
n
polygon each of which sides and diagnoals is colored with one of
n
n
n
distinct colors. For which
n
n
n
does: there exists a coloring method such that for any three of
n
n
n
colors, we can always find one triangle whose vertices is of
P
P
P
' and whose sides is colored by the three colors respectively.
1
2
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Concyclic
Given an acute triangle
P
B
C
PBC
PBC
with
P
B
≠
P
C
.
PB\neq PC.
PB
=
PC
.
Points
A
,
D
A,D
A
,
D
lie on
P
B
,
P
C
,
PB,PC,
PB
,
PC
,
respectively.
A
C
AC
A
C
intersects
B
D
BD
B
D
at point
O
.
O.
O
.
Let
E
,
F
E,F
E
,
F
be the feet of perpendiculars from
O
O
O
to
A
B
,
C
D
,
AB,CD,
A
B
,
C
D
,
respectively. Denote by
M
,
N
M,N
M
,
N
the midpoints of
B
C
,
A
D
.
BC,AD.
BC
,
A
D
.
(
1
)
(1)
(
1
)
: If four points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
lie on one circle, then EM\cdot FN \equal{} EN\cdot FM.
(
2
)
(2)
(
2
)
: Determine whether the converse of
(
1
)
(1)
(
1
)
is true or not, justify your answer.
The minimum value
Given an integer
n
>
3.
n > 3.
n
>
3.
Let
a
1
,
a
2
,
⋯
,
a
n
a_{1},a_{2},\cdots,a_{n}
a
1
,
a
2
,
⋯
,
a
n
be real numbers satisfying min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n. Find the minimum value of \sum_{k \equal{} 1}^n|a_{k}|^3.