MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2024 China National Olympiad
2024 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
Hide problems
Least swaps to get any labeling of a regular 99-gon
Let
P
P
P
be a regular
99
99
99
-gon. Assign integers between
1
1
1
and
99
99
99
to the vertices of
P
P
P
such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of
P
P
P
. Find the smallest integer
n
n
n
such that one can achieve every other assignment from a given one with no more than
n
n
n
operations.Proposed by Zhenhua Qu
5
1
Hide problems
Perfect Hexagon in China MO
In acute
△
A
B
C
\triangle {ABC}
△
A
BC
,
K
{K}
K
is on the extention of segment
B
C
BC
BC
.
P
,
Q
P, Q
P
,
Q
are two points such that
K
P
∥
A
B
,
B
K
=
B
P
KP \parallel AB, BK=BP
K
P
∥
A
B
,
B
K
=
BP
and
K
Q
∥
A
C
,
C
K
=
C
Q
KQ\parallel AC, CK=CQ
K
Q
∥
A
C
,
C
K
=
CQ
. The circumcircle of
△
K
P
Q
\triangle KPQ
△
K
PQ
intersects
A
K
AK
A
K
again at
T
{T}
T
. Prove that: (1)
∠
B
T
C
+
∠
A
P
B
=
∠
C
Q
A
\angle BTC+\angle APB=\angle CQA
∠
BTC
+
∠
A
PB
=
∠
CQ
A
. (2)
A
P
⋅
B
T
⋅
C
Q
=
A
Q
⋅
C
T
⋅
B
P
AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP
A
P
⋅
BT
⋅
CQ
=
A
Q
⋅
CT
⋅
BP
.Proposed by Yijie He and Yijuan Yao
4
1
Hide problems
Find maximum number of pairs whose product is at least 1
Let
a
1
,
a
2
,
…
,
a
2023
a_1, a_2, \ldots, a_{2023}
a
1
,
a
2
,
…
,
a
2023
be nonnegative real numbers such that
a
1
+
a
2
+
…
+
a
2023
=
100
a_1 + a_2 + \ldots + a_{2023} = 100
a
1
+
a
2
+
…
+
a
2023
=
100
. Let
A
=
{
(
i
,
j
)
∣
1
⩽
i
⩽
j
⩽
2023
,
a
i
a
j
⩾
1
}
A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}
A
=
{
(
i
,
j
)
∣
1
⩽
i
⩽
j
⩽
2023
,
a
i
a
j
⩾
1
}
. Prove that
∣
A
∣
⩽
5050
|A| \leqslant 5050
∣
A
∣
⩽
5050
and determine when the equality holds.Proposed by Yunhao Fu
1
1
Hide problems
Factorization into 2023 primes and small integers
Find the smallest
λ
∈
R
\lambda \in \mathbb{R}
λ
∈
R
such that for all
n
∈
N
+
n \in \mathbb{N}_+
n
∈
N
+
, there exists
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
satisfying
n
=
x
1
x
2
…
x
2023
n = x_1 x_2 \ldots x_{2023}
n
=
x
1
x
2
…
x
2023
, where
x
i
x_i
x
i
is either a prime or a positive integer not exceeding
n
λ
n^\lambda
n
λ
for all
i
∈
{
1
,
2
,
…
,
2023
}
i \in \left\{ 1,2, \ldots, 2023 \right\}
i
∈
{
1
,
2
,
…
,
2023
}
.Proposed by Yinghua Ai
2
1
Hide problems
China Mathematical Olympiad 2024 Q2
Find the largest real number
c
c
c
such that
∑
i
=
1
n
∑
j
=
1
n
(
n
−
∣
i
−
j
∣
)
x
i
x
j
≥
c
∑
j
=
1
n
x
i
2
\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i
i
=
1
∑
n
j
=
1
∑
n
(
n
−
∣
i
−
j
∣
)
x
i
x
j
≥
c
j
=
1
∑
n
x
i
2
for any positive integer
n
n
n
and any real numbers
x
1
,
x
2
,
…
,
x
n
.
x_1,x_2,\dots,x_n.
x
1
,
x
2
,
…
,
x
n
.
3
1
Hide problems
Including chain of minimal sets under some valuation
Let
p
⩾
5
p \geqslant 5
p
⩾
5
be a prime and
S
=
{
1
,
2
,
…
,
p
}
S = \left\{ 1, 2, \ldots, p \right\}
S
=
{
1
,
2
,
…
,
p
}
. Define
r
(
x
,
y
)
r(x,y)
r
(
x
,
y
)
as follows:
r
(
x
,
y
)
=
{
y
−
x
y
⩾
x
y
−
x
+
p
y
<
x
.
r(x,y) = \begin{cases} y - x & y \geqslant x \\ y - x + p & y < x \end{cases}.
r
(
x
,
y
)
=
{
y
−
x
y
−
x
+
p
y
⩾
x
y
<
x
.
For a nonempty proper subset
A
A
A
of
S
S
S
, let
f
(
A
)
=
∑
x
∈
A
∑
y
∈
A
(
r
(
x
,
y
)
)
2
.
f(A) = \sum_{x \in A} \sum_{y \in A} \left( r(x,y) \right)^2.
f
(
A
)
=
x
∈
A
∑
y
∈
A
∑
(
r
(
x
,
y
)
)
2
.
A good subset of
S
S
S
is a nonempty proper subset
A
A
A
satisfying that for all subsets
B
⊆
S
B \subseteq S
B
⊆
S
of the same size as
A
A
A
,
f
(
B
)
⩾
f
(
A
)
f(B) \geqslant f(A)
f
(
B
)
⩾
f
(
A
)
. Find the largest integer
L
L
L
such that there exists distinct good subsets
A
1
⊆
A
2
⊆
…
⊆
A
L
A_1 \subseteq A_2 \subseteq \ldots \subseteq A_L
A
1
⊆
A
2
⊆
…
⊆
A
L
.Proposed by Bin Wang