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Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2015 South East Mathematical Olympiad
2015 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
8
2
Hide problems
Product of two elements is an element
For any integers
m
,
n
m,n
m
,
n
, we have the set
A
(
m
,
n
)
=
{
x
2
+
m
x
+
n
∣
x
∈
Z
}
A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}
A
(
m
,
n
)
=
{
x
2
+
m
x
+
n
∣
x
∈
Z
}
, where
Z
\mathbb{Z}
Z
is the integer set. Does there exist three distinct elements
a
,
b
,
c
a,b,c
a
,
b
,
c
which belong to
A
(
m
,
n
)
A(m,n)
A
(
m
,
n
)
and satisfy the equality
a
=
b
c
a=bc
a
=
b
c
?
2015 China South East MO Grade 11 P8
Find all prime number
p
p
p
such that there exists an integer-coefficient polynomial
f
(
x
)
=
x
p
−
1
+
a
p
−
2
x
p
−
2
+
…
+
a
1
x
+
a
0
f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0
f
(
x
)
=
x
p
−
1
+
a
p
−
2
x
p
−
2
+
…
+
a
1
x
+
a
0
that has
p
−
1
p-1
p
−
1
consecutive positive integer roots and
p
2
∣
f
(
i
)
f
(
−
i
)
p^2\mid f(i)f(-i)
p
2
∣
f
(
i
)
f
(
−
i
)
, where
i
i
i
is the imaginary unit.
7
1
Hide problems
Equal product of segments
In
△
A
B
C
\triangle ABC
△
A
BC
, we have
A
B
>
A
C
>
B
C
AB>AC>BC
A
B
>
A
C
>
BC
.
D
,
E
,
F
D,E,F
D
,
E
,
F
are the tangent points of the inscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
with the line segments
A
B
,
B
C
,
A
C
AB,BC,AC
A
B
,
BC
,
A
C
respectively. The points
L
,
M
,
N
L,M,N
L
,
M
,
N
are the midpoints of the line segments
D
E
,
E
F
,
F
D
DE,EF,FD
D
E
,
EF
,
F
D
. The straight line
N
L
NL
N
L
intersects with ray
A
B
AB
A
B
at
P
P
P
, straight line
L
M
LM
L
M
intersects ray
B
C
BC
BC
at
Q
Q
Q
and the straight line
N
M
NM
NM
intersects ray
A
C
AC
A
C
at
R
R
R
. Prove that
P
A
⋅
Q
B
⋅
R
C
=
P
D
⋅
Q
E
⋅
R
F
PA \cdot QB \cdot RC = PD \cdot QE \cdot RF
P
A
⋅
QB
⋅
RC
=
P
D
⋅
QE
⋅
RF
.
6
2
Hide problems
Smallest halving cevian
In
△
A
B
C
\triangle ABC
△
A
BC
, we have three edges with lengths
B
C
=
a
,
C
A
=
b
A
B
=
c
BC=a, \, CA=b \, AB=c
BC
=
a
,
C
A
=
b
A
B
=
c
, and
c
<
b
<
a
<
2
c
c<b<a<2c
c
<
b
<
a
<
2
c
.
P
P
P
and
Q
Q
Q
are two points of the edges of
△
A
B
C
\triangle ABC
△
A
BC
, and the straight line
P
Q
PQ
PQ
divides
△
A
B
C
\triangle ABC
△
A
BC
into two parts with the same area. Find the minimum value of the length of the line segment
P
Q
PQ
PQ
.
2015 China South East MO Grade 11 P6
Given a positive integer
n
≥
2
n\geq 2
n
≥
2
. Let
A
=
{
(
a
,
b
)
∣
a
,
b
∈
{
1
,
2
,
…
,
n
}
}
A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}
A
=
{(
a
,
b
)
∣
a
,
b
∈
{
1
,
2
,
…
,
n
}}
be the set of points in Cartesian coordinate plane. How many ways to colour points in
A
A
A
, each by one of three fixed colour, such that, for any
a
,
b
∈
{
1
,
2
,
…
,
n
−
1
}
a,b\in \{ 1,2,…,n-1\}
a
,
b
∈
{
1
,
2
,
…
,
n
−
1
}
, if
(
a
,
b
)
(a,b)
(
a
,
b
)
and
(
a
+
1
,
b
)
(a+1,b)
(
a
+
1
,
b
)
have same colour, then
(
a
,
b
+
1
)
(a,b+1)
(
a
,
b
+
1
)
and
(
a
+
1
,
b
+
1
)
(a+1,b+1)
(
a
+
1
,
b
+
1
)
also have same colour.
5
2
Hide problems
Range of values
Suppose that
a
,
b
a,b
a
,
b
are real numbers, function
f
(
x
)
=
a
x
+
b
f(x) = ax+b
f
(
x
)
=
a
x
+
b
satisfies
∣
f
(
x
)
∣
≤
1
\mid f(x) \mid \leq 1
∣
f
(
x
)
∣≤
1
for any
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
. Find the range of values of
S
=
(
a
+
1
)
(
b
+
1
)
.
S= (a+1)(b+1).
S
=
(
a
+
1
)
(
b
+
1
)
.
2015 China South East MO Grade 11 P5
Given two points
E
E
E
and
F
F
F
lie on segment
A
B
AB
A
B
and
A
D
AD
A
D
, respectively. Let the segments
B
F
BF
BF
and
D
E
DE
D
E
intersects at point
C
C
C
. If it’s known that
A
E
+
E
C
=
A
F
+
F
C
AE+EC=AF+FC
A
E
+
EC
=
A
F
+
FC
, show that
A
B
+
B
C
=
A
D
+
D
C
AB+BC=AD+DC
A
B
+
BC
=
A
D
+
D
C
.
4
2
Hide problems
Lucky groups
For any positive integer
n
n
n
, we have the set
P
n
=
{
n
k
∣
k
=
0
,
1
,
2
,
…
}
P_n = \{ n^k \mid k=0,1,2, \ldots \}
P
n
=
{
n
k
∣
k
=
0
,
1
,
2
,
…
}
. For positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
, we define the group of
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
as lucky if there is a positive integer
m
m
m
such that
a
−
1
a-1
a
−
1
,
a
b
−
12
ab-12
ab
−
12
,
a
b
c
−
2015
abc-2015
ab
c
−
2015
(the three numbers need not be different from each other) belong to the set
P
m
P_m
P
m
. Find the number of lucky groups.
2015 China South East MO Grade 11 P4
Given
8
8
8
pairwise distinct positive integers
a
1
,
a
2
,
…
,
a
8
a_1,a_2,…,a_8
a
1
,
a
2
,
…
,
a
8
such that the greatest common divisor of any three of them is equal to
1
1
1
. Show that there exists positive integer
n
≥
8
n\geq 8
n
≥
8
and
n
n
n
pairwise distinct positive integers
m
1
,
m
2
,
…
,
m
n
m_1,m_2,…,m_n
m
1
,
m
2
,
…
,
m
n
with the greatest common divisor of all
n
n
n
numbers equal to
1
1
1
such that for any positive integers
1
≤
p
<
q
<
r
≤
n
1\leq p<q<r\leq n
1
≤
p
<
q
<
r
≤
n
, there exists positive integers
1
≤
i
<
j
≤
8
1\leq i<j\leq 8
1
≤
i
<
j
≤
8
that
a
i
a
j
∣
m
p
+
m
q
+
m
r
a_ia_j\mid m_p+m_q+m_r
a
i
a
j
∣
m
p
+
m
q
+
m
r
.
3
1
Hide problems
Arrange in a circle
Can you make
2015
2015
2015
positive integers
1
,
2
,
…
,
2015
1,2, \ldots , 2015
1
,
2
,
…
,
2015
to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of
4
4
4
or a multiple of
7
7
7
?
2
2
Hide problems
Incenter diamter
Let
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
. Let
Γ
\Gamma
Γ
be the circle with diameter
A
I
AI
A
I
. The circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
intersects
Γ
\Gamma
Γ
at points
A
,
D
A,D
A
,
D
, with point
D
D
D
lying on \overarc{AC} (not containing
B
B
B
). Let the line passing through
A
A
A
and parallel to
B
C
BC
BC
intersect
Γ
\Gamma
Γ
at points
A
,
E
A,E
A
,
E
. If
D
I
DI
D
I
is the angle bisector of
∠
C
D
E
\angle CDE
∠
C
D
E
, and
∠
A
B
C
=
3
3
∘
\angle ABC = 33^{\circ}
∠
A
BC
=
3
3
∘
, find the value of
∠
B
A
C
\angle BAC
∠
B
A
C
.
2015 China South East MO Grade 11 P2
Given a sequence
{
a
n
}
n
∈
Z
+
\{ a_n\}_{n\in \mathbb{Z}^+}
{
a
n
}
n
∈
Z
+
defined by
a
1
=
1
a_1=1
a
1
=
1
and
a
2
k
=
a
2
k
−
1
+
a
k
,
a
2
k
+
1
=
a
2
k
a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}
a
2
k
=
a
2
k
−
1
+
a
k
,
a
2
k
+
1
=
a
2
k
for all positive integer
k
k
k
. Prove that, for any positive integer
n
n
n
,
a
2
n
>
2
n
2
4
a_{2^n}>2^{\frac{n^2}{4}}
a
2
n
>
2
4
n
2
.
1
1
Hide problems
Sequence inequality
Suppose that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfy
a
1
=
1
a_1=1
a
1
=
1
and a_{2k}=a_{2k-1}+a_k, a_{2k+1}=a_{2k} for
k
=
1
,
2
,
…
k=1,2, \ldots
k
=
1
,
2
,
…
\\Prove that
a
2
n
<
2
n
2
2
a_{2^n}< 2^{\frac{n^2}{2}}
a
2
n
<
2
2
n
2
for any integer
n
≥
3
n \geq 3
n
≥
3
.