MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2023 China Northern MO
2023 China Northern MO
Part of
China Northern MO
Subcontests
(6)
5
1
Hide problems
combo graph inequality problem, f^ 2(G)<= c x g^3(G)
Given a finite graph
G
G
G
, let
f
(
G
)
f(G)
f
(
G
)
be the number of triangles in graph
G
G
G
,
g
(
G
)
g(G)
g
(
G
)
be the number of edges in graph
G
G
G
, find the minimum constant
c
c
c
, so that for each graph
G
G
G
, there is
f
2
(
G
)
≤
c
⋅
g
3
(
G
)
f^ 2(G)\le c \cdot g^3(G)
f
2
(
G
)
≤
c
⋅
g
3
(
G
)
.
6
1
Hide problems
infinite m = n^2+ n + 1 with pefect square factors
A positive integer
m
m
m
is called a beautiful integer if that there exists a positive integer
n
n
n
such that
m
=
n
2
+
n
+
1
m = n^2+ n + 1
m
=
n
2
+
n
+
1
. Prove that there are infinitely many beautiful integers with square factors, and the square factors of different beautiful integers are relative prime.
1
1
Hide problems
concyclic wanted, diameter related
As shown in the figure,
A
B
AB
A
B
is the diameter of circle
⊙
O
\odot O
⊙
O
, and chords
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
E
E
E
,
E
F
⊥
A
B
EF\perp AB
EF
⊥
A
B
intersects at point
F
F
F
, and
F
C
FC
FC
intersects
B
D
BD
B
D
at point
G
G
G
. Point
M
M
M
lies on
A
B
AB
A
B
such that
M
D
=
M
G
MD=MG
M
D
=
MG
. Prove that points
F
F
F
,
M
M
M
,
D
D
D
,
G
G
G
lies on a circle. https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg
3
1
Hide problems
China Northern Mathematical Olympiad 2023 , Problem 3
Find all solutions of the equation
s
i
n
π
x
+
c
o
s
π
x
=
(
−
1
)
⌊
x
⌋
sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }
s
inπ
x
+
cos
π
x
=
(
−
1
)
⌊
x
⌋
4
1
Hide problems
China Northern Mathematical Olympiad 2023 , Problem 4
Given the sequence
(
a
n
)
(a_n)
(
a
n
)
satisfies
1
=
a
1
<
a
2
<
a
3
<
⋯
<
a
n
1=a_1< a_2 < a_3< \cdots<a_n
1
=
a
1
<
a
2
<
a
3
<
⋯
<
a
n
and there exist real number
m
m
m
such that
∑
i
=
1
n
−
1
a
i
+
1
−
a
i
(
2
+
a
i
)
4
3
≤
m
\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m
i
=
1
∑
n
−
1
3
(
2
+
a
i
)
4
a
i
+
1
−
a
i
≤
m
for any positive integer
n
n
n
not less than 2 . Find the minimum of
m
.
m.
m
.
2
1
Hide problems
China Northern Mathematical Olympiad 2023 , Problem 2
Let
a
,
b
,
c
∈
(
0
,
1
)
a,b,c \in (0,1)
a
,
b
,
c
∈
(
0
,
1
)
and
a
b
+
b
c
+
c
a
=
4
a
b
c
.
ab+bc+ca=4abc .
ab
+
b
c
+
c
a
=
4
ab
c
.
Prove that
a
+
b
+
c
≥
1
−
a
+
1
−
b
+
1
−
c
\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}
a
+
b
+
c
≥
1
−
a
+
1
−
b
+
1
−
c