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Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2016 South East Mathematical Olympiad
2016 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Eventually constant
Let
{
a
n
}
\{ a_n\}
{
a
n
}
be a series consisting of positive integers such that
n
2
∣
∑
i
=
1
n
a
i
n^2 \mid \sum_{i=1}^{n}{a_i}
n
2
∣
∑
i
=
1
n
a
i
and
a
n
≤
(
n
+
2016
)
2
a_n\leq (n+2016)^2
a
n
≤
(
n
+
2016
)
2
for all
n
≥
2016
n\geq 2016
n
≥
2016
. Define
b
n
=
a
n
+
1
−
a
n
b_n=a_{n+1}-a_n
b
n
=
a
n
+
1
−
a
n
. Prove that the series
{
b
n
}
\{ b_n\}
{
b
n
}
is eventually constant.
7
2
Hide problems
Easy geometry
I
I
I
is incenter of
△
A
B
C
\triangle{ABC}
△
A
BC
. The incircle 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 . Let
M
,
N
,
K
=
B
I
,
C
I
,
D
I
∩
E
F
M,N,K=BI,CI,DI \cap EF
M
,
N
,
K
=
B
I
,
C
I
,
D
I
∩
EF
respectively and
B
N
∩
C
M
=
P
,
A
K
∩
B
C
=
G
BN\cap CM=P,AK\cap BC=G
BN
∩
CM
=
P
,
A
K
∩
BC
=
G
. Point
Q
Q
Q
is intersection of the perpendicular line to
P
G
PG
PG
through
I
I
I
and the perpendicular line to
P
B
PB
PB
through
P
P
P
. Prove that
B
I
BI
B
I
bisect segment
P
Q
PQ
PQ
.
find the value of $|P|$
Let
A
=
{
a
3
+
b
3
+
c
3
−
3
a
b
c
∣
a
,
b
,
c
∈
N
}
A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}
A
=
{
a
3
+
b
3
+
c
3
−
3
ab
c
∣
a
,
b
,
c
∈
N
}
,
B
=
{
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
∣
a
,
b
,
c
∈
N
}
B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}
B
=
{(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
∣
a
,
b
,
c
∈
N
}
,
P
=
{
n
∣
n
∈
A
∩
B
,
1
≤
n
≤
2016
}
P=\{n|n\in A\cap B,1\le n\le 2016\}
P
=
{
n
∣
n
∈
A
∩
B
,
1
≤
n
≤
2016
}
, find the value of
∣
P
∣
|P|
∣
P
∣
.
6
1
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Bad problem
Toss the coin
n
n
n
times, assume that each time, only appear only head or tail Let
a
(
n
)
a(n)
a
(
n
)
denote number of way that head appear in multiple of
3
3
3
times among
n
n
n
times Let
b
(
n
)
b(n)
b
(
n
)
denote numbe of way that head appear in multiple of
6
6
6
times among
n
n
n
times
(
1
)
(1)
(
1
)
Find
a
(
2016
)
a(2016)
a
(
2016
)
and
b
(
2016
)
b(2016)
b
(
2016
)
(
2
)
(2)
(
2
)
Find the number of positive integer
n
≤
2016
n\leq 2016
n
≤
2016
that
2
b
(
n
)
−
a
(
n
)
≥
0
2b(n)-a(n)\geq 0
2
b
(
n
)
−
a
(
n
)
≥
0
5
2
Hide problems
Easy function about divisor
Let
n
n
n
is positive integer,
D
n
D_n
D
n
is a set of all positive divisor of
n
n
n
and
f
(
n
)
=
∑
d
∈
D
n
1
1
+
d
f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}
f
(
n
)
=
∑
d
∈
D
n
1
+
d
1
Prove that for all positive integer
m
m
m
,
∑
i
=
1
m
f
(
i
)
<
m
\sum_{i=1}^{m}{f(i)} <m
∑
i
=
1
m
f
(
i
)
<
m
constant $\alpha$
Let a constant
α
\alpha
α
as
0
<
α
<
1
0<\alpha<1
0
<
α
<
1
, prove that:
(
1
)
(1)
(
1
)
There exist a constant
C
(
α
)
C(\alpha)
C
(
α
)
which is only depend on
α
\alpha
α
such that for every
x
≥
0
x\ge 0
x
≥
0
,
ln
(
1
+
x
)
≤
C
(
α
)
x
α
\ln(1+x)\le C(\alpha)x^\alpha
ln
(
1
+
x
)
≤
C
(
α
)
x
α
.
(
2
)
(2)
(
2
)
For every two complex numbers
z
1
,
z
2
z_1,z_2
z
1
,
z
2
,
∣
ln
∣
z
1
z
2
∣
∣
≤
C
(
α
)
(
∣
z
1
−
z
2
z
2
∣
α
+
∣
z
2
−
z
1
z
1
∣
α
)
|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)
∣
ln
∣
z
2
z
1
∣∣
≤
C
(
α
)
(
∣
z
2
z
1
−
z
2
∣
α
+
∣
z
1
z
2
−
z
1
∣
α
)
.
1
1
Hide problems
China South East Mathematical Olympiad 2016 Grade10 Q1
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
1
,
a
2
=
1
2
a_1=1,a_2=\frac{1}{2}
a
1
=
1
,
a
2
=
2
1
,
n
(
n
+
1
)
a
n
+
1
a
n
+
n
a
n
a
n
−
1
=
(
n
+
1
)
2
a
n
+
1
a
n
−
1
(
n
≥
2
)
.
n(n+1) a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).
n
(
n
+
1
)
a
n
+
1
a
n
+
n
a
n
a
n
−
1
=
(
n
+
1
)
2
a
n
+
1
a
n
−
1
(
n
≥
2
)
.
Prove that
2
n
+
1
<
a
n
n
<
1
n
(
n
≥
3
)
.
\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).
n
+
1
2
<
n
a
n
<
n
1
(
n
≥
3
)
.
4
2
Hide problems
Lotus design
For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of
4
4
4
points is called a "lotus design." (1) Construct an example of "lotus design". Also what are areas and distances in your example? (2) Prove that there are infinitely many "lotus design" which are not similar.
Get back to the camp
A substitute teacher lead a groop of students to go for a trip. The teacher who in charge of the groop of the students told the substitude teacher that there are two students who always lie, and the others always tell the truth. But the substitude teacher don't know who are the two students always lie. They get lost in a forest. Finally the are in a crossroad which has four roads. The substitute teacher knows that their camp is on one road, and the distence is
20
20
20
minutes' walk. The students have to go to the camp before it gets dark.
(
1
)
(1)
(
1
)
If there are
8
8
8
students, and
60
60
60
minutes before it gets dark, give a plan that all students can get back to the camp.
(
2
)
(2)
(
2
)
If there are
4
4
4
students, and
100
100
100
minutes before it gets dark, is there a plan that all students can get back to the camp?
3
1
Hide problems
n-series
Given any integer
n
≥
3
n\geq 3
n
≥
3
. A finite series is called
n
n
n
-series if it satisfies the following two conditions
1
)
1)
1
)
It has at least
3
3
3
terms and each term of it belongs to
{
1
,
2
,
.
.
.
,
n
}
\{ 1,2,...,n\}
{
1
,
2
,
...
,
n
}
2
)
2)
2
)
If series has
m
m
m
terms
a
1
,
a
2
,
.
.
.
,
a
m
a_1,a_2,...,a_m
a
1
,
a
2
,
...
,
a
m
then
(
a
k
+
1
−
a
k
)
(
a
k
+
2
−
a
k
)
<
0
(a_{k+1}-a_k)(a_{k+2}-a_k)<0
(
a
k
+
1
−
a
k
)
(
a
k
+
2
−
a
k
)
<
0
for all
k
=
1
,
2
,
.
.
.
,
m
−
2
k=1,2,...,m-2
k
=
1
,
2
,
...
,
m
−
2
How many
n
n
n
-series are there
?
?
?
2
2
Hide problems
Hard geometry
Suppose
P
A
B
PAB
P
A
B
and
P
C
D
PCD
PC
D
are two secants of circle
O
O
O
. Lines
A
D
∩
B
C
=
Q
AD \cap BC=Q
A
D
∩
BC
=
Q
. Point
T
T
T
lie on segment
B
Q
BQ
BQ
and point
K
K
K
is intersection of segment
P
T
PT
PT
with circle
O
O
O
,
S
=
Q
K
∩
P
A
S=QK\cap PA
S
=
Q
K
∩
P
A
Given that
S
T
∥
P
Q
ST \parallel PQ
ST
∥
PQ
, prove that
B
,
S
,
K
,
T
B,S,K,T
B
,
S
,
K
,
T
lie on a circle.
China South East Mathematical Olympiad 2016 Grade11 Q2
Let
n
n
n
be positive integer,
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be positive real numbers such that
x
1
x
2
⋯
x
n
=
1
x_1x_2\cdots x_n=1
x
1
x
2
⋯
x
n
=
1
. Prove that
∑
i
=
1
n
x
i
x
1
2
+
x
2
2
+
⋯
x
i
2
≥
n
+
1
2
n
\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}
i
=
1
∑
n
x
i
x
1
2
+
x
2
2
+
⋯
x
i
2
≥
2
n
+
1
n