MathDB
Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2020 South East Mathematical Olympiad
2020 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
5
1
Hide problems
Exhausting number theory
Consider the set
I
=
{
1
,
2
,
⋯
,
2020
}
I=\{ 1,2, \cdots, 2020 \}
I
=
{
1
,
2
,
⋯
,
2020
}
. Let
W
=
{
w
(
a
,
b
)
=
(
a
+
b
)
+
a
b
∣
a
,
b
∈
I
}
∩
I
W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I
W
=
{
w
(
a
,
b
)
=
(
a
+
b
)
+
ab
∣
a
,
b
∈
I
}
∩
I
,
Y
=
{
y
(
a
,
b
)
=
(
a
+
b
)
⋅
a
b
∣
a
,
b
∈
I
}
∩
I
Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I
Y
=
{
y
(
a
,
b
)
=
(
a
+
b
)
⋅
ab
∣
a
,
b
∈
I
}
∩
I
be its two subsets. Further, let
X
=
W
∩
Y
X= W \cap Y
X
=
W
∩
Y
. (1) Find the sum of maximal and minimal elements in
X
X
X
. (2) An element
n
=
y
(
a
,
b
)
(
a
≤
b
)
n=y(a,b) (a \le b)
n
=
y
(
a
,
b
)
(
a
≤
b
)
in
Y
Y
Y
is called excellent, if its representation is not unique (for instance,
20
=
y
(
1
,
5
)
=
y
(
2
,
3
)
20=y(1,5)=y(2,3)
20
=
y
(
1
,
5
)
=
y
(
2
,
3
)
). Find the number of excellent elements in
Y
Y
Y
.(2) is only for Grade 11.
2
2
Hide problems
Mixtilinear again
In a scalene triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
A
B
<
A
C
AB<AC
A
B
<
A
C
,
P
B
PB
PB
and
P
C
PC
PC
are tangents of the circumcircle
(
O
)
(O)
(
O
)
of
Δ
A
B
C
\Delta ABC
Δ
A
BC
. A point
R
R
R
lies on the arc
A
C
^
\widehat{AC}
A
C
(not containing
B
B
B
),
P
R
PR
PR
intersects
(
O
)
(O)
(
O
)
again at
Q
Q
Q
. Suppose
I
I
I
is the incenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
I
D
⊥
B
C
ID \perp BC
I
D
⊥
BC
at
D
D
D
,
Q
D
QD
Q
D
intersects
(
O
)
(O)
(
O
)
again at
G
G
G
. A line passing through
I
I
I
and perpendicular to
A
I
AI
A
I
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
M
,
N
M,N
M
,
N
, respectively. Prove that, if
A
R
∥
B
C
AR \parallel BC
A
R
∥
BC
, then
A
,
G
,
M
,
N
A,G,M,N
A
,
G
,
M
,
N
are concyclic.
Another variant of the same figure
In a scalene triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
A
B
<
A
C
AB<AC
A
B
<
A
C
,
P
B
PB
PB
and
P
C
PC
PC
are tangents of the circumcircle
(
O
)
(O)
(
O
)
of
Δ
A
B
C
\Delta ABC
Δ
A
BC
. A point
R
R
R
lies on the arc
A
C
^
\widehat{AC}
A
C
(not containing
B
B
B
),
P
R
PR
PR
intersects
(
O
)
(O)
(
O
)
again at
Q
Q
Q
. Suppose
I
I
I
is the incenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
I
D
⊥
B
C
ID \perp BC
I
D
⊥
BC
at
D
D
D
,
Q
D
QD
Q
D
intersects
(
O
)
(O)
(
O
)
again at
G
G
G
. A line passing through
I
I
I
and perpendicular to
A
I
AI
A
I
intersects
A
G
,
A
C
AG,AC
A
G
,
A
C
at
M
,
N
M,N
M
,
N
, respectively.
S
S
S
is the midpoint of arc
A
R
^
\widehat{AR}
A
R
, and
S
N
SN
SN
intersects
(
O
)
(O)
(
O
)
again at
T
T
T
. Prove that, if
A
R
∥
B
C
AR \parallel BC
A
R
∥
BC
, then
M
,
B
,
T
M,B,T
M
,
B
,
T
are collinear.
6
1
Hide problems
Pedal circle related
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
,
∠
A
B
C
=
∠
A
D
C
<
9
0
∘
\angle ABC=\angle ADC <90^{\circ}
∠
A
BC
=
∠
A
D
C
<
9
0
∘
. The circle with diameter
A
C
AC
A
C
intersects
B
C
BC
BC
and
C
D
CD
C
D
again at
E
,
F
E,F
E
,
F
, respectively.
M
M
M
is the midpoint of
B
D
BD
B
D
, and
A
N
⊥
B
D
AN \perp BD
A
N
⊥
B
D
at
N
N
N
. Prove that
M
,
N
,
E
,
F
M,N,E,F
M
,
N
,
E
,
F
is concyclic.
8
1
Hide problems
Painting with an aimless nozzle
Using a nozzle to paint each square in a
1
×
n
1 \times n
1
×
n
stripe, when the nozzle is aiming at the
i
i
i
-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of
1
2
\tfrac{1}{2}
2
1
to be painted black.In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is
T
(
n
)
T(n)
T
(
n
)
. Find the explicit formula of
T
(
n
)
T(n)
T
(
n
)
.
7
2
Hide problems
Sylvester-Schur revised
Given any prime
p
≥
3
p \ge 3
p
≥
3
. Show that for all sufficient large positive integer
x
x
x
, at least one of
x
+
1
,
x
+
2
,
⋯
,
x
+
p
+
3
2
x+1,x+2,\cdots,x+\frac{p+3}{2}
x
+
1
,
x
+
2
,
⋯
,
x
+
2
p
+
3
has a prime divisor greater than
p
p
p
.
Square-free numbers with certain gap
Arrange all square-free positive integers in ascending order
a
1
,
a
2
,
a
3
,
…
,
a
n
,
…
a_1,a_2,a_3,\ldots,a_n,\ldots
a
1
,
a
2
,
a
3
,
…
,
a
n
,
…
. Prove that there are infinitely many positive integers
n
n
n
, such that
a
n
+
1
−
a
n
=
2020
a_{n+1}-a_n=2020
a
n
+
1
−
a
n
=
2020
.
3
1
Hide problems
Polynomial without negative integer roots has many positive integer roots
Given a polynomial
f
(
x
)
=
x
2020
+
∑
i
=
0
2019
c
i
x
i
f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i
f
(
x
)
=
x
2020
+
∑
i
=
0
2019
c
i
x
i
, where
c
i
∈
{
−
1
,
0
,
1
}
c_i \in \{ -1,0,1 \}
c
i
∈
{
−
1
,
0
,
1
}
. Denote
N
N
N
the number of positive integer roots of
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
(counting multiplicity). If
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
has no negative integer roots, find the maximum of
N
N
N
.
4
2
Hide problems
2019 CSMO Grade 10 Problem 4
Let
a
1
,
a
2
,
…
,
a
17
a_1,a_2,\dots, a_{17}
a
1
,
a
2
,
…
,
a
17
be a permutation of
1
,
2
,
…
,
17
1,2,\dots, 17
1
,
2
,
…
,
17
such that
(
a
1
−
a
2
)
(
a
2
−
a
3
)
…
(
a
17
−
a
1
)
=
n
17
(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}
(
a
1
−
a
2
)
(
a
2
−
a
3
)
…
(
a
17
−
a
1
)
=
n
17
.Find the maximum possible value of
n
n
n
.
China South East Mathematical Olympiad 2020 Grade11 Q4
Let
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
−
1
≤
a
n
0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
−
1
≤
a
n
and
a
1
+
a
2
+
⋯
+
a
n
=
1.
a_1+a_2+\cdots+a_n=1.
a
1
+
a
2
+
⋯
+
a
n
=
1.
Prove that: For any non-negative numbers
x
1
,
x
2
,
⋯
,
x
n
;
y
1
,
y
2
,
⋯
,
y
n
x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n
x
1
,
x
2
,
⋯
,
x
n
;
y
1
,
y
2
,
⋯
,
y
n
, have
(
∑
i
=
1
n
a
i
x
i
−
∏
i
=
1
n
x
i
a
i
)
(
∑
i
=
1
n
a
i
y
i
−
∏
i
=
1
n
y
i
a
i
)
≤
a
n
2
(
n
∑
i
=
1
n
x
i
∑
i
=
1
n
y
i
−
∑
i
=
1
n
x
i
∑
i
=
1
n
y
i
)
2
.
\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.
(
i
=
1
∑
n
a
i
x
i
−
i
=
1
∏
n
x
i
a
i
)
(
i
=
1
∑
n
a
i
y
i
−
i
=
1
∏
n
y
i
a
i
)
≤
a
n
2
(
n
i
=
1
∑
n
x
i
i
=
1
∑
n
y
i
−
i
=
1
∑
n
x
i
i
=
1
∑
n
y
i
)
2
.
1
2
Hide problems
2019 CSMO Grade 11 Problem 1
Let
a
1
,
a
2
,
…
,
a
17
a_1,a_2,\dots, a_{17}
a
1
,
a
2
,
…
,
a
17
be a permutation of
1
,
2
,
…
,
17
1,2,\dots, 17
1
,
2
,
…
,
17
such that
(
a
1
−
a
2
)
(
a
2
−
a
3
)
…
(
a
17
−
a
1
)
=
2
n
(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n
(
a
1
−
a
2
)
(
a
2
−
a
3
)
…
(
a
17
−
a
1
)
=
2
n
. Find the maximum possible value of positive integer
n
n
n
.
China South East Mathematical Olympiad 2020 Grade10 Q1
Let
f
(
x
)
=
a
(
3
a
+
2
c
)
x
2
−
2
b
(
2
a
+
c
)
x
+
b
2
+
(
c
+
a
)
2
f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2
f
(
x
)
=
a
(
3
a
+
2
c
)
x
2
−
2
b
(
2
a
+
c
)
x
+
b
2
+
(
c
+
a
)
2
(
a
,
b
,
c
∈
R
,
a
(
3
a
+
2
c
)
≠
0
)
.
(a,b,c\in R, a(3a+2c)\neq 0).
(
a
,
b
,
c
∈
R
,
a
(
3
a
+
2
c
)
=
0
)
.
If
f
(
x
)
≤
1
f(x)\leq 1
f
(
x
)
≤
1
for any real
x
x
x
, find the maximum of
∣
a
b
∣
.
|ab|.
∣
ab
∣.