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Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2013 South East Mathematical Olympiad
2013 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
7
1
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placing angles
Given a
3
×
3
3\times 3
3
×
3
grid, we call the remainder of the grid an “angle” when a
2
×
2
2\times 2
2
×
2
grid is cut out from the grid. Now we place some angles on a
10
×
10
10\times 10
10
×
10
grid such that the borders of those angles must lie on the grid lines or its borders, moreover there is no overlap among the angles. Determine the maximal value of
k
k
k
, such that no matter how we place
k
k
k
angles on the grid, we can always place another angle on the grid.
4
1
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how many towers?
There are
12
12
12
acrobats who are assigned a distinct number (
1
,
2
,
⋯
,
12
1, 2, \cdots , 12
1
,
2
,
⋯
,
12
) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “tower” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous towers are there? (Note: two towers are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of
8
8
8
acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “towers”, however they are homogeneous towers.)
6
1
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China South East Mathematical Olympiad 2013 problem 6
n
>
1
n>1
n
>
1
is an integer. The first
n
n
n
primes are
p
1
=
2
,
p
2
=
3
,
…
,
p
n
p_1=2,p_2=3,\dotsc, p_n
p
1
=
2
,
p
2
=
3
,
…
,
p
n
. Set
A
=
p
1
p
1
p
2
p
2
.
.
.
p
n
p
n
A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}
A
=
p
1
p
1
p
2
p
2
...
p
n
p
n
. Find all positive integers
x
x
x
, such that
A
x
\dfrac Ax
x
A
is even, and
A
x
\dfrac Ax
x
A
has exactly
x
x
x
divisors
8
1
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China South East Mathematical Olympiad 2013 problem 8
n
≥
3
n\geq 3
n
≥
3
is a integer.
α
,
β
,
γ
∈
(
0
,
1
)
\alpha,\beta,\gamma \in (0,1)
α
,
β
,
γ
∈
(
0
,
1
)
. For every
a
k
,
b
k
,
c
k
≥
0
(
k
=
1
,
2
,
…
,
n
)
a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)
a
k
,
b
k
,
c
k
≥
0
(
k
=
1
,
2
,
…
,
n
)
with
∑
k
=
1
n
(
k
+
α
)
a
k
≤
α
,
∑
k
=
1
n
(
k
+
β
)
b
k
≤
β
,
∑
k
=
1
n
(
k
+
γ
)
c
k
≤
γ
\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma
k
=
1
∑
n
(
k
+
α
)
a
k
≤
α
,
k
=
1
∑
n
(
k
+
β
)
b
k
≤
β
,
k
=
1
∑
n
(
k
+
γ
)
c
k
≤
γ
, we always have
∑
k
=
1
n
(
k
+
λ
)
a
k
b
k
c
k
≤
λ
\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda
k
=
1
∑
n
(
k
+
λ
)
a
k
b
k
c
k
≤
λ
. Find the minimum of
λ
\lambda
λ
5
1
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China South East Mathematical Olympiad 2013 problem 5
f
(
x
)
=
∑
i
=
1
2013
[
x
i
!
]
f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]
f
(
x
)
=
i
=
1
∑
2013
[
i
!
x
]
. A integer
n
n
n
is called good if
f
(
x
)
=
n
f(x)=n
f
(
x
)
=
n
has real root. How many good numbers are in
{
1
,
3
,
5
,
…
,
2013
}
\{1,3,5,\dotsc,2013\}
{
1
,
3
,
5
,
…
,
2013
}
?
3
1
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China South East Mathematical Olympiad 2013 problem 3
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
,
a
1
=
1
,
a
2
=
2
,
a
n
+
1
=
a
n
2
+
(
−
1
)
n
a
n
−
1
a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}
a
1
=
1
,
a
2
=
2
,
a
n
+
1
=
a
n
−
1
a
n
2
+
(
−
1
)
n
. Show that
a
m
2
+
a
m
+
1
2
∈
{
a
n
}
,
∀
m
∈
N
a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N
a
m
2
+
a
m
+
1
2
∈
{
a
n
}
,
∀
m
∈
N
2
1
Hide problems
China South East Mathematical Olympiad 2013 problem 2
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
. the incircle
I
I
I
of
△
A
B
C
\triangle ABC
△
A
BC
meet
B
C
BC
BC
at point
D
D
D
,
A
D
AD
A
D
meet
I
I
I
again at
E
E
E
.
E
P
EP
EP
is a tangent of
I
I
I
, and
E
P
EP
EP
meet the extension line of
B
C
BC
BC
at
P
P
P
.
C
F
∥
P
E
CF\parallel PE
CF
∥
PE
,
C
F
∩
A
D
=
F
CF\cap AD=F
CF
∩
A
D
=
F
. the line
B
F
BF
BF
meet
I
I
I
at
M
,
N
M,N
M
,
N
, point
M
M
M
is on the line segment
B
F
BF
BF
, the line segment
P
M
PM
PM
meet
I
I
I
again at
Q
Q
Q
. Show that
∠
E
N
P
=
∠
E
N
Q
\angle ENP=\angle ENQ
∠
ENP
=
∠
ENQ
1
1
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China South East Mathematical Olympiad 2013 Q1
Let
a
,
b
a,b
a
,
b
be real numbers such that the equation
x
3
−
a
x
2
+
b
x
−
a
=
0
x^3-ax^2+bx-a=0
x
3
−
a
x
2
+
b
x
−
a
=
0
has three positive real roots . Find the minimum of
2
a
3
−
3
a
b
+
3
a
b
+
1
\frac{2a^3-3ab+3a}{b+1}
b
+
1
2
a
3
−
3
ab
+
3
a
.