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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2023 South East Mathematical Olympiad
2023 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
7
1
Hide problems
min max \sum a_ib_i if sum (a_i^2-b_1^2)=1 and sum (a_i+b_i)=2023
The positive integer number
S
S
S
is called a "line number". if there is a positive integer
n
n
n
and
2
n
2n
2
n
positive integers
a
1
a_1
a
1
,
a
2
a_2
a
2
,...,
a
n
a_n
a
n
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,...,
b
n
b_n
b
n
, such that
S
=
∑
i
=
1
n
a
i
b
i
S = \sum^n_{i=1} a_ib_i
S
=
∑
i
=
1
n
a
i
b
i
,
∑
i
=
1
n
(
a
i
2
−
b
1
2
)
=
1
\sum^n_{i=1} (a_i^2-b_1^2)=1
∑
i
=
1
n
(
a
i
2
−
b
1
2
)
=
1
, and
∑
i
=
1
n
(
a
i
+
b
i
)
=
2023
\sum^n_{i=1} (a_i+b_i)=2023
∑
i
=
1
n
(
a
i
+
b
i
)
=
2023
, find: (1) The minimum value of line numbers. (2)The maximum value of line numbers.
2
2
Hide problems
A division of the set of positive integers
A
A
A
is a non-empty subset of positive integers. Let
f
(
A
)
=
{
a
b
c
−
b
−
c
+
2
∣
a
,
b
,
c
∈
A
}
f(A)=\{abc-b-c+2\vert a,b,c\in A\}
f
(
A
)
=
{
ab
c
−
b
−
c
+
2∣
a
,
b
,
c
∈
A
}
Determine all integers
n
n
n
greater than
1
1
1
so that we can divide the set of positive integers into
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \dots, A_n
A
1
,
A
2
,
…
,
A
n
(
A
i
≠
∅
(
i
=
1
,
2
,
…
,
n
)
A_i\neq \emptyset (i=1, 2, \dots , n)
A
i
=
∅
(
i
=
1
,
2
,
…
,
n
)
,
∀
1
≤
i
<
j
≤
n
,
A
i
∩
A
j
=
∅
\forall 1\le i < j \le n, A_i\cap A_j = \emptyset
∀1
≤
i
<
j
≤
n
,
A
i
∩
A
j
=
∅
and
⋃
i
=
1
n
A
i
=
N
∗
\bigcup_{i=1}^{n} A_i=\mathbb{N}^*
⋃
i
=
1
n
A
i
=
N
∗
) satisfy that
∀
1
≤
i
≤
n
,
f
(
A
i
)
⊆
A
i
\forall 1\le i\le n, f(A_i) \subseteq A_i
∀1
≤
i
≤
n
,
f
(
A
i
)
⊆
A
i
.
complex number sums
For a non-empty finite complex number set
A
A
A
, define the "Tao root" of
A
A
A
as
∣
∑
z
∈
A
z
∣
\left|\sum_{z\in A} z \right|
∑
z
∈
A
z
. Given the integer
n
≥
3
n\ge 3
n
≥
3
, let the set
U
n
=
{
cos
2
k
π
n
+
i
sin
2
k
π
n
∣
k
=
0
,
1
,
.
.
.
,
n
−
1
}
.
U_n = \{\cos\frac{2k \pi}{n}+ i\sin\frac{2k \pi}{n}|k=0,1,...,n-1\}.
U
n
=
{
cos
n
2
kπ
+
i
sin
n
2
kπ
∣
k
=
0
,
1
,
...
,
n
−
1
}
.
Let
a
n
a_n
a
n
be the number of non-empty subsets in which the Tao root of
U
n
U_n
U
n
is
0
0
0
,
b
n
b_n
b
n
is the number of non-empty subsets of
U
n
U_n
U
n
whose Tao root is
1
1
1
. Compare the sizes of
n
a
n
na_n
n
a
n
and
2
b
n
2b_n
2
b
n
.
5
2
Hide problems
The intersection lies on a circle
Let
A
B
AB
A
B
be a chord of the semicircle
O
O
O
(not the diameter).
M
M
M
is the midpoint of
A
B
AB
A
B
, and
D
D
D
is a point lies on line
O
M
OM
OM
(
D
D
D
is outside semicircle
O
O
O
). Line
l
l
l
passes through
D
D
D
and is parallel to
A
B
AB
A
B
.
P
,
Q
P, Q
P
,
Q
are two points lie on
l
l
l
and
P
O
PO
PO
meets semicircle
O
O
O
at
C
C
C
. If
∠
P
C
D
=
∠
D
M
C
\angle PCD=\angle DMC
∠
PC
D
=
∠
D
MC
, and
M
M
M
is the orthocentre of
△
O
P
Q
\triangle OPQ
△
OPQ
. Prove that the intersection of
A
Q
AQ
A
Q
and
P
B
PB
PB
lies on semicircle
O
O
O
.
concyclic wanted, touchpoints of incircle
As shown in the figure, in
△
A
B
C
\vartriangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
, the inscribed circle
I
I
I
is tangent to the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at points
D
D
D
,
E
E
E
,
F
F
F
respectively, and the straight lines
B
C
BC
BC
and
E
F
EF
EF
intersect at point
K
K
K
,
D
G
⊥
E
F
DG \perp EF
D
G
⊥
EF
at point
G
G
G
, ray
I
G
IG
I
G
intersects the circumscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
at point
H
H
H
. Prove that points
H
H
H
,
G
G
G
,
D
D
D
,
K
K
K
lie on a circle. https://cdn.artofproblemsolving.com/attachments/5/e/804fb919e9c2f9cf612099e44bad9c75699b2e.png
8
2
Hide problems
p(x) has at most n-1 different rational roots
Let
p
(
x
)
p(x)
p
(
x
)
be an
n
n
n
-degree
(
n
≥
2
)
(n \ge 2)
(
n
≥
2
)
polynomial with integer coefficients. If there are infinitely many positive integers
m
m
m
, such that
p
(
m
)
p(m)
p
(
m
)
at most
n
−
1
n -1
n
−
1
different prime factors
f
f
f
, prove that
p
(
x
)
p(x)
p
(
x
)
has at most
n
−
1
n-1
n
−
1
different rational roots .[color=#f00]a help in translation is welcome
Game strategy with flipping coins
Let
n
{n}
n
be a fixed positive integer.
A
{A}
A
and
B
{B}
B
play the following game:
2023
2023
2023
coins marked
1
,
2
,
…
,
2023
1, 2, \dots, 2023
1
,
2
,
…
,
2023
lie on a circle (the marks are considered in module
2023
2023
2023
) and each coin has two sides. Initially, all coins are head up and
A
{A}
A
's goal is to make as many coins with tail up. In each operation,
A
{A}
A
choose two coins marked
k
{k}
k
and
k
+
3
k+3
k
+
3
with head up (if
A
{A}
A
can't choose, the game ends) and
B
{B}
B
choose a coin marked
k
+
1
k+1
k
+
1
or
k
+
2
k+2
k
+
2
and flip it. If at some moment there are
n
{n}
n
coins with tail up,
A
{A}
A
wins. Find the largest
n
{n}
n
such that
A
{A}
A
has a winning strategy.
6
2
Hide problems
China South East Mathematical Olympiad 2023 Grade10 Q6
Let
a
1
≥
a
2
≥
⋯
≥
a
n
>
0.
a_1\geq a_2\geq \cdots \geq a_n >0 .
a
1
≥
a
2
≥
⋯
≥
a
n
>
0.
Prove that
(
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
)
2
≥
∑
k
=
1
n
k
(
2
k
−
1
)
a
1
2
+
a
2
2
+
⋯
+
a
k
2
\left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}
(
a
1
1
+
a
2
1
+
...
+
a
n
1
)
2
≥
k
=
1
∑
n
a
1
2
+
a
2
2
+
⋯
+
a
k
2
k
(
2
k
−
1
)
P_n(x) = T(P_{n-1}(x))
Let
R
[
x
]
R[x]
R
[
x
]
be the whole set of real coefficient polynomials, and define the mapping
T
:
R
[
x
]
→
R
[
x
]
T: R[x] \to R[x]
T
:
R
[
x
]
→
R
[
x
]
as follows: For
f
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
,
f (x) = a_nx^{n} + a_{n-1}x^{n- 1} +...+ a_1x + a_0,
f
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
,
let
T
(
f
(
x
)
)
=
a
n
x
n
+
1
+
a
n
−
1
x
n
+
(
a
n
+
a
n
−
2
)
x
n
−
1
+
(
a
n
−
1
+
a
n
−
3
)
x
n
−
2
+
.
.
.
+
(
a
2
+
a
0
)
x
+
a
1
.
T(f(x))=a_{n}x^{n+1} + a_{n-1}x^{n} + (a_n+a_{n-2})x^{n-1 } + (a_{n-1}+a_{n-3})x^{n-2}+...+(a_2+a_0)x+a_1.
T
(
f
(
x
))
=
a
n
x
n
+
1
+
a
n
−
1
x
n
+
(
a
n
+
a
n
−
2
)
x
n
−
1
+
(
a
n
−
1
+
a
n
−
3
)
x
n
−
2
+
...
+
(
a
2
+
a
0
)
x
+
a
1
.
Assume
P
0
(
x
)
=
1
P_0(x)= 1
P
0
(
x
)
=
1
,
P
n
(
x
)
=
T
(
P
n
−
1
(
x
)
)
P_n(x) = T(P_{n-1}(x))
P
n
(
x
)
=
T
(
P
n
−
1
(
x
))
(
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
), find the constant term of
P
n
(
x
)
P_n(x)
P
n
(
x
)
.
1
2
Hide problems
An easy problem about a sequence
The positive sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies:
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
2
+
a
n
−
1
−
2
1
+
a
n
−
1
(
n
≥
2
)
a_n=2+\sqrt{a_{n-1}}-2 \sqrt{1+\sqrt{a_{n-1}}}(n\geq 2)
a
n
=
2
+
a
n
−
1
−
2
1
+
a
n
−
1
(
n
≥
2
)
Let
S
n
=
∑
k
=
1
n
2
k
a
k
S_n=\sum\limits_{k=1}^{n}{2^ka_k}
S
n
=
k
=
1
∑
n
2
k
a
k
. Find the value of
S
2023
S_{2023}
S
2023
.
Easy 2-variable inequality
Let
a
,
b
>
0
a, b>0
a
,
b
>
0
. Prove that:
(
a
3
+
b
3
+
a
3
b
3
)
(
1
a
3
+
1
b
3
+
1
a
3
b
3
)
+
27
≥
6
(
a
+
b
+
1
a
+
1
b
+
a
b
+
b
a
)
(a^3+b^3+a^3b^3)(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{a^3b^3} ) +27 \ge 6(a+b+\frac{1}{a} +\frac{1}{b} +\frac{a}{b} +\frac{b}{a})
(
a
3
+
b
3
+
a
3
b
3
)
(
a
3
1
+
b
3
1
+
a
3
b
3
1
)
+
27
≥
6
(
a
+
b
+
a
1
+
b
1
+
b
a
+
a
b
)
3
2
Hide problems
Prove collinear
In acute triangle
A
B
C
ABC
A
BC
(
△
A
B
C
\triangle ABC
△
A
BC
is not an isosceles triangle),
I
I
I
is its incentre, and circle
ω
\omega
ω
is its inscribed circle.
⊙
ω
\odot\omega
⊙
ω
touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively.
A
D
AD
A
D
intersects with
⊙
ω
\odot\omega
⊙
ω
at
J
J
J
(
J
≠
D
J\neq D
J
=
D
), and the circumcircle of
△
B
C
J
\triangle BCJ
△
BC
J
intersects
⊙
ω
\odot\omega
⊙
ω
at
K
K
K
(
K
≠
J
K\neq J
K
=
J
). The circumcircle of
△
B
F
K
\triangle BFK
△
BF
K
and
△
C
E
K
\triangle CEK
△
CE
K
meet at
L
L
L
(
L
≠
K
L\neq K
L
=
K
). Let
M
M
M
be the midpoint of the major arc
B
A
C
BAC
B
A
C
. Prove that
M
,
I
,
L
M, I, L
M
,
I
,
L
are collinear.
Beautiful conditional geo in China Southeast
In
△
A
B
C
\triangle {ABC}
△
A
BC
,
D
{D}
D
is on the internal angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and
∠
A
D
B
=
∠
A
C
D
\angle ADB=\angle ACD
∠
A
D
B
=
∠
A
C
D
.
E
,
F
E, F
E
,
F
is on the external angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
, such that
A
E
=
B
E
AE=BE
A
E
=
BE
and
A
F
=
C
F
AF=CF
A
F
=
CF
. The circumcircles of
△
A
C
E
\triangle ACE
△
A
CE
and
△
A
B
F
\triangle ABF
△
A
BF
intersects at
A
{A}
A
and
K
{K}
K
and
A
′
A'
A
′
is the reflection of
A
{A}
A
with respect to
B
C
BC
BC
. Prove that: if
A
D
=
B
C
AD=BC
A
D
=
BC
, then the circumcenter of
△
A
K
A
′
\triangle AKA'
△
A
K
A
′
is on line
A
D
AD
A
D
.
4
2
Hide problems
Inequality of decimal part
Find the largest real number
c
c
c
, such that for any integer
s
>
1
s>1
s
>
1
, and positive integers
m
,
n
m, n
m
,
n
coprime to
s
s
s
, we have
∑
j
=
1
s
−
1
{
j
m
s
}
(
1
−
{
j
m
s
}
)
{
j
n
s
}
(
1
−
{
j
n
s
}
)
≥
c
s
\sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs
j
=
1
∑
s
−
1
{
s
jm
}
(
1
−
{
s
jm
})
{
s
jn
}
(
1
−
{
s
jn
})
≥
cs
where
{
x
}
=
x
−
⌊
x
⌋
\{ x \} = x - \lfloor x \rfloor
{
x
}
=
x
−
⌊
x
⌋
.
Pingsheng Number
Given an integer
n
≥
2
n\geq 2
n
≥
2
. Call a positive integer
T
{T}
T
Pingsheng Number, if there exists pairwise different non empty subsets
A
1
,
A
2
,
⋯
,
A
m
A_1,A_2,\cdots ,A_m
A
1
,
A
2
,
⋯
,
A
m
(
m
≥
3
)
(m\geq 3)
(
m
≥
3
)
of set
S
=
{
1
,
2
,
⋯
,
n
}
,
S=\{1,2,\cdots ,n\},
S
=
{
1
,
2
,
⋯
,
n
}
,
satisfying
T
=
∑
i
=
1
m
∣
A
i
∣
,
T=\sum\limits_{i=1}^m|A_i|,
T
=
i
=
1
∑
m
∣
A
i
∣
,
and for
∀
p
,
q
,
r
∈
{
1
,
2
,
⋯
,
m
}
,
p
≠
q
,
q
≠
r
,
r
≠
p
,
\forall p,q,r\in\{1,2,\cdots ,m\},p\neq q,q\neq r,r\neq p,
∀
p
,
q
,
r
∈
{
1
,
2
,
⋯
,
m
}
,
p
=
q
,
q
=
r
,
r
=
p
,
we have
A
p
∩
(
A
q
△
A
r
)
=
∅
A_p\cap(A_q\triangle A_r)=\varnothing
A
p
∩
(
A
q
△
A
r
)
=
∅
or
A
p
⊆
(
A
q
△
A
r
)
.
A_p\subseteq (A_q\triangle A_r).
A
p
⊆
(
A
q
△
A
r
)
.
Find the max Pingsheng Number.