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Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2023 China National Olympiad
2023 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
5
1
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China MO 2023 P5
Prove that there exist
C
>
0
C>0
C
>
0
, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence
a
1
,
a
2
,
a
3
,
⋯
a_1, a_2, a_3,\cdots
a
1
,
a
2
,
a
3
,
⋯
, if the greatest common divisor of
a
1
a_1
a
1
and
a
2
a_2
a
2
is squarefree, then there exists a positive integer
m
≤
C
⋅
a
2
2
m\le C\cdot {a_2}^2
m
≤
C
⋅
a
2
2
, such that
a
m
a_m
a
m
is squarefree. Note: A positive integer
N
N
N
is squarefree if it is not divisible by any square number greater than
1
1
1
.Proposed by Qu Zhenhua
6
1
Hide problems
China MO 2023 P6
There are
n
(
n
≥
8
)
n(n\ge 8)
n
(
n
≥
8
)
airports, some of which have one-way direct routes between them. For any two airports
a
a
a
and
b
b
b
, there is at most one one-way direct route from
a
a
a
to
b
b
b
(there may be both one-way direct routes from
a
a
a
to
b
b
b
and from
b
b
b
to
a
a
a
). For any set
A
A
A
composed of airports
(
1
≤
∣
A
∣
≤
n
−
1
)
(1\le | A| \le n-1)
(
1
≤
∣
A
∣
≤
n
−
1
)
, there are at least
4
⋅
min
{
∣
A
∣
,
n
−
∣
A
∣
}
4\cdot \min \{|A|,n-|A| \}
4
⋅
min
{
∣
A
∣
,
n
−
∣
A
∣
}
one-way direct routes from the airport in
A
A
A
to the airport not in
A
A
A
. Prove that: For any airport
x
x
x
, we can start from
x
x
x
and return to the airport by no more than
2
n
\sqrt{2n}
2
n
one-way direct routes.
4
1
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China MO 2023 P4
Find the minimum positive integer
n
≥
3
n\ge 3
n
≥
3
, such that there exist
n
n
n
points
A
1
,
A
2
,
⋯
,
A
n
A_1,A_2,\cdots, A_n
A
1
,
A
2
,
⋯
,
A
n
satisfying no three points are collinear and for any
1
≤
i
≤
n
1\le i\le n
1
≤
i
≤
n
, there exist
1
≤
j
≤
n
(
j
≠
i
)
1\le j \le n (j\neq i)
1
≤
j
≤
n
(
j
=
i
)
, segment
A
j
A
j
+
1
A_jA_{j+1}
A
j
A
j
+
1
pass through the midpoint of segment
A
i
A
i
+
1
A_iA_{i+1}
A
i
A
i
+
1
, where
A
n
+
1
=
A
1
A_{n+1}=A_1
A
n
+
1
=
A
1
2
1
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Orz Orz Orz Orz Orz geometry
Let
△
A
B
C
\triangle ABC
△
A
BC
be an equilateral triangle of side length 1. Let
D
,
E
,
F
D,E,F
D
,
E
,
F
be points on
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively, such that
D
E
20
=
E
F
22
=
F
D
38
\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}
20
D
E
=
22
EF
=
38
F
D
. Let
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
be on lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively, such that
X
Y
⊥
D
E
,
Y
Z
⊥
E
F
,
Z
X
⊥
F
D
XY\perp DE, YZ\perp EF, ZX\perp FD
X
Y
⊥
D
E
,
Y
Z
⊥
EF
,
ZX
⊥
F
D
. Find all possible values of
1
[
D
E
F
]
+
1
[
X
Y
Z
]
\frac{1}{[DEF]} + \frac{1}{[XYZ]}
[
D
EF
]
1
+
[
X
Y
Z
]
1
.
3
1
Hide problems
China MO 2023 P3
Given positive integer
m
,
n
m,n
m
,
n
, color the points of the regular
(
2
m
+
2
n
)
(2m+2n)
(
2
m
+
2
n
)
-gon in black and white,
2
m
2m
2
m
in black and
2
n
2n
2
n
in white. The coloring distance
d
(
B
,
C
)
d(B,C)
d
(
B
,
C
)
of two black points
B
,
C
B,C
B
,
C
is defined as the smaller number of white points in the two paths linking the two black points. The coloring distance
d
(
W
,
X
)
d(W,X)
d
(
W
,
X
)
of two white points
W
,
X
W,X
W
,
X
is defined as the smaller number of black points in the two paths linking the two white points. We define the matching of black points
B
\mathcal{B}
B
: label the
2
m
2m
2
m
black points with
A
1
,
⋯
,
A
m
,
B
1
,
⋯
,
B
m
A_1,\cdots,A_m,B_1,\cdots,B_m
A
1
,
⋯
,
A
m
,
B
1
,
⋯
,
B
m
satisfying no
A
i
B
i
A_iB_i
A
i
B
i
intersects inside the gon. We define the matching of white points
W
\mathcal{W}
W
: label the
2
n
2n
2
n
white points with
C
1
,
⋯
,
C
n
,
D
1
,
⋯
,
D
n
C_1,\cdots,C_n,D_1,\cdots,D_n
C
1
,
⋯
,
C
n
,
D
1
,
⋯
,
D
n
satisfying no
C
i
D
i
C_iD_i
C
i
D
i
intersects inside the gon. We define
P
(
B
)
=
∑
i
=
1
m
d
(
A
i
,
B
i
)
,
P
(
W
)
=
∑
j
=
1
n
d
(
C
j
,
D
j
)
P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j)
P
(
B
)
=
∑
i
=
1
m
d
(
A
i
,
B
i
)
,
P
(
W
)
=
∑
j
=
1
n
d
(
C
j
,
D
j
)
. Prove that:
max
B
P
(
B
)
=
max
W
P
(
W
)
\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})
max
B
P
(
B
)
=
max
W
P
(
W
)
1
1
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China MO 2023 Problem 1: Brother sequences
Define the sequences
(
a
n
)
,
(
b
n
)
(a_n),(b_n)
(
a
n
)
,
(
b
n
)
by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If
a
100
b
100
=
a
101
b
101
a_{100}b_{100} = a_{101}b_{101}
a
100
b
100
=
a
101
b
101
, find the value of
a
1
−
b
1
a_1-b_1
a
1
−
b
1
; 2) If
a
100
=
b
99
a_{100} = b_{99}
a
100
=
b
99
, determine which is larger between
a
100
+
b
100
a_{100}+b_{100}
a
100
+
b
100
and
a
101
+
b
101
a_{101}+b_{101}
a
101
+
b
101
.