MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2024 China Second Round
2024 China Second Round
Part of
(China) National High School Mathematics League
Subcontests
(4)
1
1
Hide problems
the nearest distance in geometric sequence
A positive integer
r
r
r
is given, find the largest real number
C
C
C
such that there exists a geometric sequence
{
a
n
}
n
≥
1
\{ a_n \}_{n\ge 1}
{
a
n
}
n
≥
1
with common ratio
r
r
r
satisfying
∥
a
n
∥
≥
C
\| a_n \| \ge C
∥
a
n
∥
≥
C
for all positive integers
n
n
n
. Here,
∥
x
∥
\| x \|
∥
x
∥
denotes the distance from the real number
x
x
x
to the nearest integer.
2
1
Hide problems
Geometry in 2024 China High School Olympics
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral,
A
C
AC
A
C
bisects the angle
∠
B
A
D
\angle BAD
∠
B
A
D
. Points
E
E
E
and
F
F
F
are on the sides
B
C
BC
BC
and
C
D
CD
C
D
respectively such that
E
F
∥
B
D
EF \parallel BD
EF
∥
B
D
. Extend
F
A
FA
F
A
and
E
A
EA
E
A
to points
P
P
P
and
Q
Q
Q
respectively, such that the circle
ω
1
\omega_1
ω
1
passing through points
A
A
A
,
B
B
B
,
P
P
P
and the circle
ω
2
\omega_2
ω
2
passing through points
A
A
A
,
D
D
D
,
Q
Q
Q
are both tangent to line
A
C
AC
A
C
. Prove that the points
B
B
B
,
P
P
P
,
Q
Q
Q
,
D
D
D
are concyclic.
4
1
Hide problems
all coprime integers included
Let
A
A
A
and
B
B
B
be positive integers, and let
S
S
S
be a set of positive integers with the following properties: (1) For every non-negative integer
k
k
k
,
A
k
∈
S
\text{ } A^k \in S
A
k
∈
S
; (2) If a positive integer
n
∈
S
n \in S
n
∈
S
, then every positive divisor of
n
n
n
is in
S
S
S
; (3) If
m
,
n
∈
S
m ,n \in S
m
,
n
∈
S
and
m
,
n
m,n
m
,
n
are coprime, then
m
n
∈
S
mn \in S
mn
∈
S
; (4) If
n
∈
S
n \in S
n
∈
S
, then
A
n
+
B
∈
S
An + B \in S
A
n
+
B
∈
S
. Prove that all positive integers coprime to
B
B
B
are in
S
S
S
.
3
1
Hide problems
connected set in grid
Given a positive integer
n
n
n
. Consider a
3
×
n
3 \times n
3
×
n
grid, a set
S
S
S
of squares is called connected if for any points
A
≠
B
A \neq B
A
=
B
in
S
S
S
, there exists an integer
l
≥
2
l \ge 2
l
≥
2
and
l
l
l
squares
A
=
C
1
,
C
2
,
…
,
C
l
=
B
A=C_1,C_2,\dots ,C_l=B
A
=
C
1
,
C
2
,
…
,
C
l
=
B
in
S
S
S
such that
C
i
C_i
C
i
and
C
i
+
1
C_{i+1}
C
i
+
1
shares a common side (
i
=
1
,
2
,
…
,
l
−
1
i=1,2,\dots,l-1
i
=
1
,
2
,
…
,
l
−
1
). Find the largest integer
K
K
K
satisfying that however the squares are colored black or white, there always exists a connected set
S
S
S
for which the absolute value of the difference between the number of black and white squares is at least
K
K
K
.