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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2008 South East Mathematical Olympiad
2008 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(4)
4
2
Hide problems
derived sequence
Let
m
,
n
m, n
m
,
n
be positive integers
(
m
,
n
>
=
2
)
(m, n>=2)
(
m
,
n
>=
2
)
. Given an
n
n
n
-element set
A
A
A
of integers
(
A
=
{
a
1
,
a
2
,
⋯
,
a
n
}
)
(A=\{a_1,a_2,\cdots ,a_n\})
(
A
=
{
a
1
,
a
2
,
⋯
,
a
n
})
, for each pair of elements
a
i
,
a
j
(
j
>
i
)
a_i, a_j(j>i)
a
i
,
a
j
(
j
>
i
)
, we make a difference by
a
j
−
a
i
a_j-a_i
a
j
−
a
i
. All these
C
n
2
C^2_n
C
n
2
differences form an ascending sequence called “derived sequence” of set
A
A
A
. Let
A
ˉ
\bar{A}
A
ˉ
denote the derived sequence of set
A
A
A
. Let
A
ˉ
(
m
)
\bar{A}(m)
A
ˉ
(
m
)
denote the number of terms divisible by
m
m
m
in
A
ˉ
\bar{A}
A
ˉ
. Prove that
A
ˉ
(
m
)
≥
B
ˉ
(
m
)
\bar{A}(m)\ge \bar{B}(m)
A
ˉ
(
m
)
≥
B
ˉ
(
m
)
where
A
=
{
a
1
,
a
2
,
⋯
,
a
n
}
A=\{a_1,a_2,\cdots ,a_n\}
A
=
{
a
1
,
a
2
,
⋯
,
a
n
}
and
B
=
{
1
,
2
,
⋯
,
n
}
B=\{1,2,\cdots ,n\}
B
=
{
1
,
2
,
⋯
,
n
}
.
wave numbers
Let
n
n
n
be a positive integer.
f
(
n
)
f(n)
f
(
n
)
denotes the number of
n
n
n
-digit numbers
a
1
a
2
⋯
a
n
‾
\overline{a_1a_2\cdots a_n}
a
1
a
2
⋯
a
n
(wave numbers) satisfying the following conditions : (i) for each
a
i
∈
{
1
,
2
,
3
,
4
}
a_i \in\{1,2,3,4\}
a
i
∈
{
1
,
2
,
3
,
4
}
,
a
i
≠
a
i
+
1
a_i \not= a_{i+1}
a
i
=
a
i
+
1
,
i
=
1
,
2
,
⋯
i=1,2,\cdots
i
=
1
,
2
,
⋯
; (ii) for
n
≥
3
n\ge 3
n
≥
3
,
(
a
i
−
a
i
+
1
)
(
a
i
+
1
−
a
i
+
2
)
(a_i-a_{i+1})(a_{i+1}-a_{i+2})
(
a
i
−
a
i
+
1
)
(
a
i
+
1
−
a
i
+
2
)
is negative,
i
=
1
,
2
,
⋯
i=1,2,\cdots
i
=
1
,
2
,
⋯
. (1) Find the value of
f
(
10
)
f(10)
f
(
10
)
; (2) Determine the remainder of
f
(
2008
)
f(2008)
f
(
2008
)
upon division by
13
13
13
.
3
2
Hide problems
four points lie on a circle
In
△
A
B
C
\triangle ABC
△
A
BC
, side
B
C
>
A
B
BC>AB
BC
>
A
B
. Point
D
D
D
lies on side
A
C
AC
A
C
such that
∠
A
B
D
=
∠
C
B
D
\angle ABD=\angle CBD
∠
A
B
D
=
∠
CB
D
. Points
Q
,
P
Q,P
Q
,
P
lie on line
B
D
BD
B
D
such that
A
Q
⊥
B
D
AQ\bot BD
A
Q
⊥
B
D
and
C
P
⊥
B
D
CP\bot BD
CP
⊥
B
D
.
M
,
E
M,E
M
,
E
are the midpoints of side
A
C
AC
A
C
and
B
C
BC
BC
respectively. Circle
O
O
O
is the circumcircle of
△
P
Q
M
\triangle PQM
△
PQM
intersecting side
A
C
AC
A
C
at
H
H
H
. Prove that
O
,
H
,
E
,
M
O,H,E,M
O
,
H
,
E
,
M
lie on a circle.
pirates and treasure chests
Captain Jack and his pirate men plundered six chests of treasure
(
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
)
(A_1,A_2,A_3,A_4,A_5,A_6)
(
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
)
. Every chest
A
i
A_i
A
i
contains
a
i
a_i
a
i
coins of gold, and all
a
i
a_i
a
i
s are pairwise different
(
i
=
1
,
2
,
⋯
,
6
)
(i=1,2,\cdots ,6)
(
i
=
1
,
2
,
⋯
,
6
)
. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?
1
2
Hide problems
set and subset
Given a set
S
=
{
1
,
2
,
3
,
…
,
3
n
}
,
(
n
∈
N
∗
)
S=\{1,2,3,\ldots,3n\},(n\in N^*)
S
=
{
1
,
2
,
3
,
…
,
3
n
}
,
(
n
∈
N
∗
)
, let
T
T
T
be a subset of
S
S
S
, such that for any
x
,
y
,
z
∈
T
x, y, z\in T
x
,
y
,
z
∈
T
(not necessarily distinct) we have
x
+
y
+
z
∉
T
x+y+z\not \in T
x
+
y
+
z
∈
T
. Find the maximum number of elements
T
T
T
can have.
find the maximal value
Let
λ
\lambda
λ
be a positive real number. Inequality
∣
λ
x
y
+
y
z
∣
≤
5
2
|\lambda xy+yz|\le \dfrac{\sqrt5}{2}
∣
λ
x
y
+
yz
∣
≤
2
5
holds for arbitrary real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying
x
2
+
y
2
+
z
2
=
1
x^2+y^2+z^2=1
x
2
+
y
2
+
z
2
=
1
. Find the maximal value of
λ
\lambda
λ
.
2
2
Hide problems
find general term formula for sequence
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence satisfying:
a
1
=
1
a_1=1
a
1
=
1
and
a
n
+
1
=
2
a
n
+
n
⋅
(
1
+
2
n
)
,
(
n
=
1
,
2
,
3
,
⋯
)
a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)
a
n
+
1
=
2
a
n
+
n
⋅
(
1
+
2
n
)
,
(
n
=
1
,
2
,
3
,
⋯
)
. Determine the general term formula of
{
a
n
}
\{a_n\}
{
a
n
}
.
three points are collinear
Circle
I
I
I
is the incircle of
△
A
B
C
\triangle ABC
△
A
BC
. Circle
I
I
I
is tangent to sides
B
C
BC
BC
and
A
C
AC
A
C
at
M
,
N
M,N
M
,
N
respectively.
E
,
F
E,F
E
,
F
are midpoints of sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Lines
E
F
,
B
I
EF, BI
EF
,
B
I
intersect at
D
D
D
. Show that
M
,
N
,
D
M,N,D
M
,
N
,
D
are collinear.