MathDB
Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2019 South East Mathematical Olympiad
2019 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
8
2
Hide problems
Long Problem I
For positive integer
x
>
1
x>1
x
>
1
, define set
S
x
S_x
S
x
as
S
x
=
{
p
α
∣
p
is one of the prime divisor of
x
,
α
∈
N
,
p
α
∣
x
,
α
≡
v
p
(
x
)
(
mod
2
)
}
,
S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},
S
x
=
{
p
α
∣
p
is one of the prime divisor of
x
,
α
∈
N
,
p
α
∣
x
,
α
≡
v
p
(
x
)
(
mod
2
)}
,
where
v
p
(
n
)
v_p(n)
v
p
(
n
)
is the power of prime divisor
p
p
p
in positive integer
n
.
n.
n
.
Let
f
(
x
)
f(x)
f
(
x
)
be the sum of all the elements of
S
x
S_x
S
x
when
x
>
1
,
x>1,
x
>
1
,
and
f
(
1
)
=
1.
f(1)=1.
f
(
1
)
=
1.
Let
m
m
m
be a given positive integer, and the sequence
a
1
,
a
2
,
⋯
,
a
n
,
⋯
a_1,a_2,\cdots,a_n,\cdots
a
1
,
a
2
,
⋯
,
a
n
,
⋯
satisfy that for any positive integer
n
>
m
,
n>m,
n
>
m
,
a
n
+
1
=
max
{
f
(
a
n
)
,
f
(
a
n
−
1
+
1
)
,
⋯
,
f
(
a
n
−
m
+
m
)
}
.
a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.
a
n
+
1
=
max
{
f
(
a
n
)
,
f
(
a
n
−
1
+
1
)
,
⋯
,
f
(
a
n
−
m
+
m
)}
.
Prove that (1)there exists constant
A
,
B
(
0
<
A
<
1
)
,
A,B(0<A<1),
A
,
B
(
0
<
A
<
1
)
,
such that when positive integer
x
x
x
has at least two different prime divisors,
f
(
x
)
<
A
x
+
B
f(x)<Ax+B
f
(
x
)
<
A
x
+
B
holds; (2)there exists positive integer
Q
Q
Q
, such that for any positive integer
n
,
a
n
<
Q
.
n,a_n<Q.
n
,
a
n
<
Q
.
Long Problem II
For positive integer
x
>
1
x>1
x
>
1
, define set
S
x
S_x
S
x
as
S
x
=
{
p
α
∣
p
is one of the prime divisor of
x
,
α
∈
N
,
p
α
∣
x
,
α
≡
v
p
(
x
)
(
mod
2
)
}
,
S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\},
S
x
=
{
p
α
∣
p
is one of the prime divisor of
x
,
α
∈
N
,
p
α
∣
x
,
α
≡
v
p
(
x
)
(
mod
2
)}
,
where
v
p
(
n
)
v_p(n)
v
p
(
n
)
is the power of prime divisor
p
p
p
in positive integer
n
.
n.
n
.
Let
f
(
x
)
f(x)
f
(
x
)
be the sum of all the elements of
S
x
S_x
S
x
when
x
>
1
,
x>1,
x
>
1
,
and
f
(
1
)
=
1.
f(1)=1.
f
(
1
)
=
1.
Let
m
m
m
be a given positive integer, and the sequence
a
1
,
a
2
,
⋯
,
a
n
,
⋯
a_1,a_2,\cdots,a_n,\cdots
a
1
,
a
2
,
⋯
,
a
n
,
⋯
satisfy that for any positive integer
n
>
m
,
n>m,
n
>
m
,
a
n
+
1
=
max
{
f
(
a
n
)
,
f
(
a
n
−
1
+
1
)
,
⋯
,
f
(
a
n
−
m
+
m
)
}
.
a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}.
a
n
+
1
=
max
{
f
(
a
n
)
,
f
(
a
n
−
1
+
1
)
,
⋯
,
f
(
a
n
−
m
+
m
)}
.
Prove that (1)there exists constant
A
,
B
(
0
<
A
<
1
)
,
A,B(0<A<1),
A
,
B
(
0
<
A
<
1
)
,
such that when positive integer
x
x
x
has at least two different prime divisors,
f
(
x
)
<
A
x
+
B
f(x)<Ax+B
f
(
x
)
<
A
x
+
B
holds; (2)there exists positive integer
N
,
l
N,l
N
,
l
, such that for any positive integer
n
≥
N
,
a
n
+
l
=
a
n
n\geq N ,a_{n+l}=a_n
n
≥
N
,
a
n
+
l
=
a
n
holds.
5
2
Hide problems
Red Subset?
Let
S
=
{
1928
,
1929
,
1930
,
⋯
,
1949
}
.
S=\{1928,1929,1930,\cdots,1949\}.
S
=
{
1928
,
1929
,
1930
,
⋯
,
1949
}
.
We call one of
S
S
S
’s subset
M
M
M
is a red subset, if the sum of any two different elements of
M
M
M
isn’t divided by
4.
4.
4.
Let
x
,
y
x,y
x
,
y
be the number of the red subsets of
S
S
S
with
4
4
4
and
5
5
5
elements,respectively. Determine which of
x
,
y
x,y
x
,
y
is greater and prove that.
One sequence problem
For positive integer n, define
a
n
a_n
a
n
as the number of the triangles with integer length of every side and the length of the longest side being
2
n
.
2n.
2
n
.
(1) Find
a
n
a_n
a
n
in terms of
n
;
n;
n
;
(2)If the sequence
{
b
n
}
\{ b_n\}
{
b
n
}
satisfying for any positive integer
n
,
n,
n
,
∑
k
=
1
n
(
−
1
)
n
−
k
(
n
k
)
b
k
=
a
n
.
\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n.
∑
k
=
1
n
(
−
1
)
n
−
k
(
k
n
)
b
k
=
a
n
.
Find the number of positive integer
n
n
n
satisfying that
b
n
≤
2019
a
n
.
b_n\leq 2019a_n.
b
n
≤
2019
a
n
.
7
2
Hide problems
Strange Geometry
Let
A
B
C
D
ABCD
A
BC
D
be a given convex quadrilateral in a plane. Prove that there exist a line with four different points
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
on it and a square
A
’
B
’
C
’
D
’
A’B’C’D’
A
’
B
’
C
’
D
’
such that
P
P
P
lies on both line
A
B
AB
A
B
and
A
’
B
’
,
A’B’,
A
’
B
’
,
Q
Q
Q
lies on both line
B
C
BC
BC
and
B
’
C
’
,
B’C’,
B
’
C
’
,
R
R
R
lies on both line
C
D
CD
C
D
and
C
’
D
’
,
C’D’,
C
’
D
’
,
S
S
S
lies on both line
D
A
DA
D
A
and
D
’
A
’
.
D’A’.
D
’
A
’.
choose numbers from 0,1,2,...,81
Amy and Bob choose numbers from
0
,
1
,
2
,
⋯
,
81
0,1,2,\cdots,81
0
,
1
,
2
,
⋯
,
81
in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all
82
82
82
numbers are chosen, let
A
A
A
be the sum of all the numbers Amy chooses, and let
B
B
B
be the sum of all the numbers Bob chooses. During the process, Amy tries to make
gcd
(
A
,
B
)
\gcd(A,B)
g
cd
(
A
,
B
)
as great as possible, and Bob tries to make
gcd
(
A
,
B
)
\gcd(A,B)
g
cd
(
A
,
B
)
as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine
gcd
(
A
,
B
)
\gcd(A,B)
g
cd
(
A
,
B
)
when all
82
82
82
numbers are chosen.
6
2
Hide problems
the maximum of axy+byz+czx
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the lengths of the sides of a given triangle.If positive reals
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy
x
+
y
+
z
=
1
,
x+y+z=1,
x
+
y
+
z
=
1
,
find the maximum of
a
x
y
+
b
y
z
+
c
z
x
.
axy+byz+czx.
a
x
y
+
b
yz
+
cz
x
.
Angle bisectors and parallel
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
, the bisectors of
∠
A
B
C
,
∠
A
C
B
\angle ABC, \angle ACB
∠
A
BC
,
∠
A
CB
meet sides
A
C
,
A
B
AC,AB
A
C
,
A
B
at
D
,
E
D,E
D
,
E
respectively. The tangent at
A
A
A
to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
intersects
E
D
ED
E
D
extended at
P
P
P
. Suppose that
A
P
=
B
C
AP=BC
A
P
=
BC
. Prove that
B
D
∥
C
P
BD\parallel CP
B
D
∥
CP
.
4
2
Hide problems
Banners cover grid table
As the figure is shown, place a
2
×
5
2\times 5
2
×
5
grid table in horizontal or vertical direction, and then remove arbitrary one
1
×
1
1\times 1
1
×
1
square on its four corners. The eight different shapes consisting of the remaining nine small squares are called banners. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed
9
×
18
9\times 18
9
×
18
grid table. Find the number of ways to cover the grid table completely with 18 banners.
Nasty 5x5 table
Let
X
X
X
be a
5
×
5
5\times 5
5
×
5
matrix with each entry be
0
0
0
or
1
1
1
. Let
x
i
,
j
x_{i,j}
x
i
,
j
be the
(
i
,
j
)
(i,j)
(
i
,
j
)
-th entry of
X
X
X
(i,j=1,2,\hdots,5). Consider all the
24
24
24
ordered sequence in the rows, columns and diagonals of
X
X
X
in the following: \begin{align*} &(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\ &(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\ &(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\ &(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5}) \end{align*} Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in
X
X
X
.
1
2
Hide problems
Non-homogeneous and asymmetric
Find the largest real number
k
k
k
, such that for any positive real numbers
a
,
b
a,b
a
,
b
,
(
a
+
b
)
(
a
b
+
1
)
(
b
+
1
)
≥
k
a
b
2
(a+b)(ab+1)(b+1)\geq kab^2
(
a
+
b
)
(
ab
+
1
)
(
b
+
1
)
≥
ka
b
2
Does there exists sequence satisfying (mod 10) relation?
Let
[
a
]
[a]
[
a
]
represent the largest integer less than or equal to
a
a
a
, for any real number
a
a
a
. Let
{
a
}
=
a
−
[
a
]
\{a\} = a - [a]
{
a
}
=
a
−
[
a
]
.Are there positive integers
m
,
n
m,n
m
,
n
and
n
+
1
n+1
n
+
1
real numbers x_0,x_1,\hdots,x_n such that
x
0
=
428
x_0=428
x
0
=
428
,
x
n
=
1928
x_n=1928
x
n
=
1928
,
x
k
+
1
10
=
[
x
k
10
]
+
m
+
{
x
k
5
}
\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\}
10
x
k
+
1
=
[
10
x
k
]
+
m
+
{
5
x
k
}
holds?Justify your answer.
2
2
Hide problems
Two circles are congruent if tangent
Two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at
A
,
B
A,B
A
,
B
. Points
C
,
D
C,D
C
,
D
lie on
Γ
1
\Gamma_1
Γ
1
, points
E
,
F
E,F
E
,
F
lie on
Γ
2
\Gamma_2
Γ
2
such that
A
,
B
A,B
A
,
B
lies on segments
C
E
,
D
F
CE,DF
CE
,
D
F
respectively and segments
C
E
,
D
F
CE,DF
CE
,
D
F
do not intersect. Let
C
F
CF
CF
meet
Γ
1
,
Γ
2
\Gamma_1,\Gamma_2
Γ
1
,
Γ
2
again at
K
,
L
K,L
K
,
L
respectively, and
D
E
DE
D
E
meet
Γ
1
,
Γ
2
\Gamma_1,\Gamma_2
Γ
1
,
Γ
2
at
M
,
N
M,N
M
,
N
respectively. If the circumcircles of
△
A
L
M
\triangle ALM
△
A
L
M
and
△
B
K
N
\triangle BKN
△
B
K
N
are tangent, prove that the radii of these two circles are equal.
Concyclic with China
A
B
C
D
ABCD
A
BC
D
is a parallelogram with
∠
B
A
D
≠
90
\angle BAD \neq 90
∠
B
A
D
=
90
. Circle centered at
A
A
A
radius
B
A
BA
B
A
denoted as
ω
1
\omega _1
ω
1
intersects the extended side of
A
B
,
C
B
AB,CB
A
B
,
CB
at points
E
,
F
E,F
E
,
F
respectively. Suppose the circle centered at
D
D
D
with radius
D
A
DA
D
A
, denoted as
ω
2
\omega _2
ω
2
, intersects
A
D
,
C
D
AD,CD
A
D
,
C
D
at points
M
,
N
M,N
M
,
N
respectively. Suppose
E
N
,
F
M
EN,FM
EN
,
FM
intersects at
G
G
G
,
A
G
AG
A
G
intersects
M
E
ME
ME
at point
T
T
T
.
M
F
MF
MF
intersects
ω
1
\omega _1
ω
1
at
Q
≠
F
Q \neq F
Q
=
F
, and
E
N
EN
EN
intersects
ω
2
\omega _2
ω
2
at
P
≠
N
P \neq N
P
=
N
. Prove that
G
,
P
,
T
,
Q
G,P,T,Q
G
,
P
,
T
,
Q
concyclic.
3
2
Hide problems
Infinitely many k such that f(k)=1
Let
f
:
N
→
N
f:\mathbb{N}\rightarrow \mathbb{N}
f
:
N
→
N
be a function such that
f
(
a
b
)
f(ab)
f
(
ab
)
divides
max
{
f
(
a
)
,
b
}
\max \{f(a),b\}
max
{
f
(
a
)
,
b
}
for any positive integers
a
,
b
a,b
a
,
b
. Must there exist infinitely many positive integers
k
k
k
such that
f
(
k
)
=
1
f(k)=1
f
(
k
)
=
1
?
Cut the perfect square out
n
n
n
symbols line up in a row, numbered as
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
from left to right. Delete every symbol with squared numbers. Renumber the rest from left to right. Repeat the process until all
n
n
n
symbols are deleted. Let
f
(
n
)
f(n)
f
(
n
)
be the initial number of the last symbol deleted. Find
f
(
n
)
f(n)
f
(
n
)
in terms of
n
n
n
and find
f
(
2019
)
f(2019)
f
(
2019
)
.