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Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2015 China Western Mathematical Olympiad
2015 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(8)
8
1
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Fermat prime
Let
k
k
k
be a positive integer, and
n
=
(
2
k
)
!
n=\left(2^k\right)!
n
=
(
2
k
)
!
.Prove that
σ
(
n
)
\sigma(n)
σ
(
n
)
has at least a prime divisor larger than
2
k
2^k
2
k
, where
σ
(
n
)
\sigma(n)
σ
(
n
)
is the sum of all positive divisors of
n
n
n
.
7
1
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Geometric inequality with complex numbers
Let
a
∈
(
0
,
1
)
a\in (0,1)
a
∈
(
0
,
1
)
,
f
(
z
)
=
z
2
−
z
+
a
,
z
∈
C
f(z)=z^2-z+a, z\in \mathbb{C}
f
(
z
)
=
z
2
−
z
+
a
,
z
∈
C
. Prove the following statement holds: For any complex number z with
∣
z
∣
≥
1
|z| \geq 1
∣
z
∣
≥
1
, there exists a complex number
z
0
z_0
z
0
with
∣
z
0
∣
=
1
|z_0|=1
∣
z
0
∣
=
1
, such that
∣
f
(
z
0
)
∣
≤
∣
f
(
z
)
∣
|f(z_0)| \leq |f(z)|
∣
f
(
z
0
)
∣
≤
∣
f
(
z
)
∣
.
6
1
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Arithmetic progression in sets
For a sequence
a
1
,
a
2
,
.
.
.
,
a
m
a_1,a_2,...,a_m
a
1
,
a
2
,
...
,
a
m
of real numbers, define the following sets
A
=
{
a
i
∣
1
≤
i
≤
m
}
and
B
=
{
a
i
+
2
a
j
∣
1
≤
i
,
j
≤
m
,
i
≠
j
}
A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}
A
=
{
a
i
∣1
≤
i
≤
m
}
and
B
=
{
a
i
+
2
a
j
∣1
≤
i
,
j
≤
m
,
i
=
j
}
Let
n
n
n
be a given integer, and
n
>
2
n>2
n
>
2
. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set
A
△
B
A\triangle B
A
△
B
, where
A
△
B
A\triangle B
A
△
B
=
(
A
∪
B
)
∖
(
A
∩
B
)
.
= \left(A\cup B\right) \setminus \left(A\cap B\right).
=
(
A
∪
B
)
∖
(
A
∩
B
)
.
4
1
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Set of right-angled triangles
For
100
100
100
straight lines on a plane, let
T
T
T
be the set of all right-angled triangles bounded by some
3
3
3
lines. Determine, with proof, the maximum value of
∣
T
∣
|T|
∣
T
∣
.
2
1
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A property of touching circles
Two circles
(
Ω
1
)
,
(
Ω
2
)
\left(\Omega_1\right),\left(\Omega_2\right)
(
Ω
1
)
,
(
Ω
2
)
touch internally on the point
T
T
T
. Let
M
,
N
M,N
M
,
N
be two points on the circle
(
Ω
1
)
\left(\Omega_1\right)
(
Ω
1
)
which are different from
T
T
T
and
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four points on
(
Ω
2
)
\left(\Omega_2\right)
(
Ω
2
)
such that the chords
A
B
,
C
D
AB, CD
A
B
,
C
D
pass through
M
,
N
M,N
M
,
N
, respectively. Prove that if
A
C
,
B
D
,
M
N
AC,BD,MN
A
C
,
B
D
,
MN
have a common point
K
K
K
, then
T
K
TK
T
K
is the angle bisector of
∠
M
T
N
\angle MTN
∠
MTN
.*
(
Ω
2
)
\left(\Omega_2\right)
(
Ω
2
)
is bigger than
(
Ω
1
)
\left(\Omega_1\right)
(
Ω
1
)
5
1
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China Western Mathematical Olympiad 2015 ,Problem 5
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are lengths of the sides of a convex quadrangle with the area equal to
S
S
S
, set
S
=
{
x
1
,
x
2
,
x
3
,
x
4
}
S =\{x_1, x_2,x_3,x_4\}
S
=
{
x
1
,
x
2
,
x
3
,
x
4
}
consists of permutations
x
i
x_i
x
i
of
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
. Prove that
S
≤
1
2
(
x
1
x
2
+
x
3
x
4
)
.
S \leq \frac{1}{2}(x_1x_2+x_3x_4).
S
≤
2
1
(
x
1
x
2
+
x
3
x
4
)
.
1
1
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China Western Mathematical Olympiad 2015 ,Problem 1
Let the integer
n
≥
2
n \ge 2
n
≥
2
, and
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be real numbers such that
∑
k
=
1
n
x
k
\sum_{k=1}^nx_k
∑
k
=
1
n
x
k
be integer .
d
k
=
min
m
∈
Z
∣
x
k
−
m
∣
d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right|
d
k
=
m
∈
Z
min
∣
x
k
−
m
∣
,
1
≤
k
≤
n
1\leq k\leq n
1
≤
k
≤
n
.Find the maximum value of
∑
k
=
1
n
d
k
\sum_{k=1}^nd_k
∑
k
=
1
n
d
k
.
3
1
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China Western Mathematical Olympiad 2015 ,Problem 3
Let the integer
n
≥
2
n \ge 2
n
≥
2
, and
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be positive real numbers such that
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^nx_i=1
∑
i
=
1
n
x
i
=
1
.Prove that
(
∑
i
=
1
n
1
1
−
x
i
)
(
∑
1
≤
i
<
j
≤
n
x
i
x
j
)
≤
n
2
.
\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.
(
i
=
1
∑
n
1
−
x
i
1
)
(
1
≤
i
<
j
≤
n
∑
x
i
x
j
)
≤
2
n
.