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Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2019 China Western Mathematical Olympiad
2019 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(8)
5
1
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2019 China Western Mathematical Olympiad Q5
In acute-angled triangle
A
B
C
,
ABC,
A
BC
,
A
B
>
A
C
.
AB>AC.
A
B
>
A
C
.
Let
O
,
H
O,H
O
,
H
be the circumcenter and orthocenter of
△
A
B
C
,
\triangle ABC,
△
A
BC
,
respectively. The line passing through
H
H
H
and parallel to
A
B
AB
A
B
intersects line
A
C
AC
A
C
at
M
,
M,
M
,
and the line passing through
H
H
H
and parallel to
A
C
AC
A
C
intersects line
A
B
AB
A
B
at
N
.
N.
N
.
L
L
L
is the reflection of the point
H
H
H
in
M
N
.
MN.
MN
.
Line
O
L
OL
O
L
and
A
H
AH
A
H
intersect at
K
.
K.
K
.
Prove that
K
,
M
,
L
,
N
K,M,L,N
K
,
M
,
L
,
N
are concyclic.
7
1
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2019 China Western Mathematical Olympiad Q7
Prove that for any positive integer
k
,
k,
k
,
there exist finitely many sets
T
T
T
satisfying the following two properties:
(
1
)
T
(1)T
(
1
)
T
consists of finitely many prime numbers;
(
2
)
∏
p
∈
T
(
p
+
k
)
(2)\textup{ }\prod_{p\in T} (p+k)
(
2
)
∏
p
∈
T
(
p
+
k
)
is divisible by
∏
p
∈
T
p
.
\prod_{p\in T} p.
∏
p
∈
T
p
.
8
1
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{x,2x,3x} is a good set
We call a set
S
S
S
a good set if
S
=
{
x
,
2
x
,
3
x
}
(
x
≠
0
)
.
S=\{x,2x,3x\}(x\neq 0).
S
=
{
x
,
2
x
,
3
x
}
(
x
=
0
)
.
For a given integer
n
(
n
≥
3
)
,
n(n\geq 3),
n
(
n
≥
3
)
,
determine the largest possible number of the good subsets of a set containing
n
n
n
positive integers.
6
1
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2019 China Western Mathematical Olympiad Q6
Let
a
1
,
a
2
,
⋯
,
a
n
(
n
≥
2
)
a_1,a_2,\cdots,a_n (n\ge 2)
a
1
,
a
2
,
⋯
,
a
n
(
n
≥
2
)
be positive numbers such that
a
1
≤
a
2
≤
⋯
≤
a
n
.
a_1\leq a_2 \leq \cdots \leq a_n .
a
1
≤
a
2
≤
⋯
≤
a
n
.
Prove that
∑
1
≤
i
<
j
≤
n
(
a
i
+
a
j
)
2
(
1
i
2
+
1
j
2
)
≥
4
(
n
−
1
)
∑
i
=
1
n
a
i
2
i
2
.
\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.
1
≤
i
<
j
≤
n
∑
(
a
i
+
a
j
)
2
(
i
2
1
+
j
2
1
)
≥
4
(
n
−
1
)
i
=
1
∑
n
i
2
a
i
2
.
3
1
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Maximal rectangles with different coloured corners
Let
S
=
{
(
i
,
j
)
∣
i
,
j
=
1
,
2
,
…
,
100
}
S=\{(i,j) \vert i,j=1,2,\ldots ,100\}
S
=
{(
i
,
j
)
∣
i
,
j
=
1
,
2
,
…
,
100
}
be a set consisting of points on the coordinate plane. Each element of
S
S
S
is colored one of four given colors. A subset
T
T
T
of
S
S
S
is called colorful if
T
T
T
consists of exactly
4
4
4
points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets
S
S
S
can have, among all legitimate coloring patters.
4
1
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Sum of distances to nearest integer
Let
n
n
n
be a given integer such that
n
≥
2
n\ge 2
n
≥
2
. Find the smallest real number
λ
\lambda
λ
with the following property: for any real numbers
x
1
,
x
2
,
…
,
x
n
∈
[
0
,
1
]
x_1,x_2,\ldots ,x_n\in [0,1]
x
1
,
x
2
,
…
,
x
n
∈
[
0
,
1
]
, there exists integers
ε
1
,
ε
2
,
…
,
ε
n
∈
{
0
,
1
}
\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}
ε
1
,
ε
2
,
…
,
ε
n
∈
{
0
,
1
}
such that the inequality
∣
∑
k
=
i
j
(
ε
k
−
x
k
)
∣
≤
λ
\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda
k
=
i
∑
j
(
ε
k
−
x
k
)
≤
λ
holds for all pairs of integers
(
i
,
j
)
(i,j)
(
i
,
j
)
where
1
≤
i
≤
j
≤
n
1\le i\le j\le n
1
≤
i
≤
j
≤
n
.
2
1
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Symmedian perpendicular to Euler line
Let
O
,
H
O,H
O
,
H
be the circumcenter and orthocenter of acute triangle
A
B
C
ABC
A
BC
with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
, respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
K
K
K
be the intersection of
A
M
AM
A
M
and the circumcircle of
△
B
H
C
\triangle BHC
△
B
H
C
, such that
M
M
M
lies between
A
A
A
and
K
K
K
. Let
N
N
N
be the intersection of
H
K
HK
HK
and
B
C
BC
BC
. Show that if
∠
B
A
M
=
∠
C
A
N
\angle BAM=\angle CAN
∠
B
A
M
=
∠
C
A
N
, then
A
N
⊥
O
H
AN\perp OH
A
N
⊥
O
H
.
1
1
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3^n+n^2+2019 is a perfect square
Determine all the possible positive integer
n
,
n,
n
,
such that
3
n
+
n
2
+
2019
3^n+n^2+2019
3
n
+
n
2
+
2019
is a perfect square.