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Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2024 China Western Mathematical Olympiad
2024 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(8)
8
1
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Maximum value of n^2 reals in a grid
Given a positive integer
n
≥
2
n \geq 2
n
≥
2
. Let
a
i
j
a_{ij}
a
ij
(
1
≤
i
,
j
≤
n
)
(1 \leq i,j \leq n)
(
1
≤
i
,
j
≤
n
)
be
n
2
n^2
n
2
non-negative reals and their sum is
1
1
1
. For
1
≤
i
≤
n
1\leq i \leq n
1
≤
i
≤
n
, define
R
i
=
m
a
x
1
≤
k
≤
n
(
a
i
k
)
R_i=max_{1\leq k \leq n}(a_{ik})
R
i
=
ma
x
1
≤
k
≤
n
(
a
ik
)
. For
1
≤
j
≤
n
1\leq j \leq n
1
≤
j
≤
n
, define
C
j
=
m
i
n
1
≤
k
≤
n
(
a
k
j
)
C_j=min_{1\leq k \leq n}(a_{kj})
C
j
=
mi
n
1
≤
k
≤
n
(
a
kj
)
Find the maximum value of
C
1
C
2
⋯
C
n
(
R
1
+
R
2
+
⋯
+
R
n
)
C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)
C
1
C
2
⋯
C
n
(
R
1
+
R
2
+
⋯
+
R
n
)
7
1
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Divisible problem
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be four positive integers such that
a
>
b
>
c
>
d
a>b>c>d
a
>
b
>
c
>
d
. Given that
a
b
+
b
c
+
c
a
+
d
2
∣
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
ab+bc+ca+d^2|(a+b)(b+c)(c+a)
ab
+
b
c
+
c
a
+
d
2
∣
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
. Find the minimal value of
Ω
(
a
b
+
b
c
+
c
a
+
d
2
)
\Omega (ab+bc+ca+d^2)
Ω
(
ab
+
b
c
+
c
a
+
d
2
)
. Here
Ω
(
n
)
\Omega(n)
Ω
(
n
)
denotes the number of prime factors
n
n
n
has. e.g.
Ω
(
12
)
=
3
\Omega(12)=3
Ω
(
12
)
=
3
6
1
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Alice and her rabbit
Alice and Bob now play a magic show. There are
101
101
101
different hats lie on the table and they form a circle. Firstly, Bob choose a positive integer
n
n
n
(Alice doesn’t know it). Then Bob puts a rabbit under one of the hats and Alice doesn’t know which hat contains the rabbit. Each time, she can choose a hat and see whether the rabbit is under the hat. If not, then Bob will move the rabbit from the current hat to the
n
n
n
th hat in a clockwise direction. They will repeat these steps until Alice find the rabbit. Prove that Alice can find the rabbit in
201
201
201
steps.
5
1
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Hexagon in a square
Given hexagon
P
\mathcal{P}
P
inscribed in a unit square, such that each vertex is on the side of the square. It’s known that all interior angles of the hexagon are equal. Find the maximum possible value of the smallest side length of
P
\mathcal{P}
P
.
4
1
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Finding good pairs
Given positive integer
n
≥
2
n \geq 2
n
≥
2
. Now Cindy fills each cell of the
n
∗
n
n*n
n
∗
n
grid with a positive integer not greater than
n
n
n
such that the numbers in each row are in a non-decreasing order (from left to right) and numbers in each column is also in a non-decreasing order (from top to bottom). We call two adjacant cells form a
d
o
m
i
n
o
domino
d
o
min
o
, if they are filled with the same number. Now Cindy wants the number of
d
o
m
i
n
o
domino
d
o
min
o
s as small as possible. Find the smallest number of
d
o
m
i
n
o
s
dominos
d
o
min
os
Cindy can reach. (Here, two cells are called adjacant if they share one common side)
3
1
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Tangent and circles with equal angles
A
B
,
A
C
AB,AC
A
B
,
A
C
are tangent to
Ω
\Omega
Ω
at
B
B
B
and
C
C
C
, respectively.
D
,
E
,
F
D,E,F
D
,
E
,
F
lie on segments
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
such that
A
F
<
A
E
AF<AE
A
F
<
A
E
and
∠
F
D
B
=
∠
E
D
C
\angle FDB= \angle EDC
∠
F
D
B
=
∠
E
D
C
. The circumcircle of
△
F
E
C
\triangle FEC
△
FEC
intersects
Ω
\Omega
Ω
at
G
G
G
and
C
C
C
. Show that
∠
A
E
F
=
∠
B
G
D
\angle AEF= \angle BGD
∠
A
EF
=
∠
BG
D
2
1
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Integer sequence with bounds
Find all integers
k
k
k
, such that there exists an integer sequence
{
a
n
}
{\{a_n\}}
{
a
n
}
satisfies two conditions below (1) For all positive integers
n
n
n
,
a
n
+
1
=
a
n
3
+
k
a
n
+
1
a_{n+1}={a_n}^3+ka_n+1
a
n
+
1
=
a
n
3
+
k
a
n
+
1
(2)
∣
a
n
∣
≤
M
|a_n| \leq M
∣
a
n
∣
≤
M
holds for some real
M
M
M
1
1
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Sum of squares in 1865
For positive integer
n
n
n
, note
S
n
=
1
2024
+
2
2024
+
⋯
+
n
2024
S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}
S
n
=
1
2024
+
2
2024
+
⋯
+
n
2024
. Prove that there exists infinitely many positive integers
n
n
n
, such that
S
n
S_n
S
n
isn’t divisible by
1865
1865
1865
but
S
n
+
1
S_{n+1}
S
n
+
1
is divisible by
1865
1865
1865