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Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2011 China Western Mathematical Olympiad
2011 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(4)
4
2
Hide problems
Tangent
In a circle
Γ
1
\Gamma_{1}
Γ
1
, centered at
O
O
O
,
A
B
AB
A
B
and
C
D
CD
C
D
are two unequal in length chords intersecting at
E
E
E
inside
Γ
1
\Gamma_{1}
Γ
1
. A circle
Γ
2
\Gamma_{2}
Γ
2
, centered at
I
I
I
is tangent to
Γ
1
\Gamma_{1}
Γ
1
internally at
F
F
F
, and also tangent to
A
B
AB
A
B
at
G
G
G
and
C
D
CD
C
D
at
H
H
H
. A line
l
l
l
through
O
O
O
intersects
A
B
AB
A
B
and
C
D
CD
C
D
at
P
P
P
and
Q
Q
Q
respectively such that
E
P
=
E
Q
EP = EQ
EP
=
EQ
. The line
E
F
EF
EF
intersects
l
l
l
at
M
M
M
. Prove that the line through
M
M
M
parallel to
A
B
AB
A
B
is tangent to
Γ
1
\Gamma_{1}
Γ
1
divisibility
Find all pairs of integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
n
∣
(
a
n
+
b
n
+
1
)
n|( a^n + b^{n+1})
n
∣
(
a
n
+
b
n
+
1
)
for all positive integer
n
n
n
2
2
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Maximum size
Let
M
M
M
be a subset of
{
1
,
2
,
3...2011
}
\{1,2,3... 2011\}
{
1
,
2
,
3...2011
}
satisfying the following condition: For any three elements in
M
M
M
, there exist two of them
a
a
a
and
b
b
b
such that
a
∣
b
a|b
a
∣
b
or
b
∣
a
b|a
b
∣
a
. Determine the maximum value of
∣
M
∣
|M|
∣
M
∣
where
∣
M
∣
|M|
∣
M
∣
denotes the number of elements in
M
M
M
Cauchy-like ineq
Let
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
, prove that
(
a
−
b
)
2
(
c
+
a
)
(
c
+
b
)
+
(
b
−
c
)
2
(
a
+
b
)
(
a
+
c
)
+
(
c
−
a
)
2
(
b
+
c
)
(
b
+
a
)
≥
(
a
−
b
)
2
a
2
+
b
2
+
c
2
\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}
(
c
+
a
)
(
c
+
b
)
(
a
−
b
)
2
+
(
a
+
b
)
(
a
+
c
)
(
b
−
c
)
2
+
(
b
+
c
)
(
b
+
a
)
(
c
−
a
)
2
≥
a
2
+
b
2
+
c
2
(
a
−
b
)
2
1
2
Hide problems
Maximum value
Given that
0
<
x
,
y
<
1
0 < x,y < 1
0
<
x
,
y
<
1
, find the maximum value of
x
y
(
1
−
x
−
y
)
(
x
+
y
)
(
1
−
x
)
(
1
−
y
)
\frac{xy(1-x-y)}{(x+y)(1-x)(1-y)}
(
x
+
y
)
(
1
−
x
)
(
1
−
y
)
x
y
(
1
−
x
−
y
)
perfect squares
Does there exist any odd integer
n
≥
3
n \geq 3
n
≥
3
and
n
n
n
distinct prime numbers
p
1
,
p
2
,
⋯
p
n
p_1 , p_2, \cdots p_n
p
1
,
p
2
,
⋯
p
n
such that all
p
i
+
p
i
+
1
(
i
=
1
,
2
,
⋯
,
n
p_i + p_{i+1} (i = 1,2,\cdots , n
p
i
+
p
i
+
1
(
i
=
1
,
2
,
⋯
,
n
and
p
n
+
1
=
p
1
)
p_{n+1} = p_{1})
p
n
+
1
=
p
1
)
are perfect squares?
3
2
Hide problems
Arranging subsets
Let
n
≥
2
n \geq 2
n
≥
2
be a given integer
a
)
a)
a
)
Prove that one can arrange all the subsets of the set
{
1
,
2...
,
n
}
\{1,2... ,n\}
{
1
,
2...
,
n
}
as a sequence of subsets
A
1
,
A
2
,
⋯
,
A
2
n
A_{1}, A_{2},\cdots , A_{2^{n}}
A
1
,
A
2
,
⋯
,
A
2
n
, such that
∣
A
i
+
1
∣
=
∣
A
i
∣
+
1
|A_{i+1}| = |A_{i}| + 1
∣
A
i
+
1
∣
=
∣
A
i
∣
+
1
or
∣
A
i
∣
−
1
|A_{i}| - 1
∣
A
i
∣
−
1
where
i
=
1
,
2
,
3
,
⋯
,
2
n
i = 1,2,3,\cdots , 2^{n}
i
=
1
,
2
,
3
,
⋯
,
2
n
and
A
2
n
+
1
=
A
1
A_{2^{n} + 1} = A_{1}
A
2
n
+
1
=
A
1
b
)
b)
b
)
Determine all possible values of the sum
∑
i
=
1
2
n
(
−
1
)
i
S
(
A
i
)
\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})
i
=
1
∑
2
n
(
−
1
)
i
S
(
A
i
)
where
S
(
A
i
)
S(A_{i})
S
(
A
i
)
denotes the sum of all elements in
A
i
A_{i}
A
i
and
S
(
∅
)
=
0
S(\emptyset) = 0
S
(
∅
)
=
0
, for any subset sequence
A
1
,
A
2
,
⋯
,
A
2
n
A_{1},A_{2},\cdots ,A_{2^n}
A
1
,
A
2
,
⋯
,
A
2
n
satisfying the condition in
a
)
a)
a
)
Concyclic Points
In triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
and incenter
I
I
I
, the incircle touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively.
M
M
M
is the midpoint of
B
C
BC
BC
, and the altitude at
A
A
A
meets
B
C
BC
BC
at
H
H
H
. Ray
A
I
AI
A
I
meets lines
D
E
DE
D
E
and
D
F
DF
D
F
at
K
K
K
and
L
L
L
, respectively. Prove that the points
M
,
L
,
H
,
K
M,L,H,K
M
,
L
,
H
,
K
are concyclic.