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National and Regional Contests
China Contests
South East Mathematical Olympiad
2017 South East Mathematical Olympiad
2017 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(8)
7
2
Hide problems
CSMO Grade 10 Problem 7
Let
m
m
m
be a given positive integer. Define
a
k
=
(
2
k
m
)
!
3
(
k
−
1
)
m
,
k
=
1
,
2
,
⋯
.
a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.
a
k
=
3
(
k
−
1
)
m
(
2
km
)!
,
k
=
1
,
2
,
⋯
.
Prove that there are infinite many integers and infinite many non-integers in the sequence
{
a
k
}
\{a_k\}
{
a
k
}
.
CSMO Grade 11 Problem 7
Find the maximum value of
n
n
n
, such that there exist
n
n
n
pairwise distinct positive numbers
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
, satisfy
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
2017
x_1^2+x_2^2+\cdots+x_n^2=2017
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
2017
8
2
Hide problems
CSMO Grade 10 Problem 8
Given the positive integer
m
≥
2
m \geq 2
m
≥
2
,
n
≥
3
n \geq 3
n
≥
3
. Define the following set
S
=
{
(
a
,
b
)
∣
a
∈
{
1
,
2
,
⋯
,
m
}
,
b
∈
{
1
,
2
,
⋯
,
n
}
}
.
S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.
S
=
{
(
a
,
b
)
∣
a
∈
{
1
,
2
,
⋯
,
m
}
,
b
∈
{
1
,
2
,
⋯
,
n
}
}
.
Let
A
A
A
be a subset of
S
S
S
. If there does not exist positive integers
x
1
,
x
2
,
y
1
,
y
2
,
y
3
x_1, x_2, y_1, y_2, y_3
x
1
,
x
2
,
y
1
,
y
2
,
y
3
such that
x
1
<
x
2
,
y
1
<
y
2
<
y
3
x_1 < x_2, y_1 < y_2 < y_3
x
1
<
x
2
,
y
1
<
y
2
<
y
3
and
(
x
1
,
y
1
)
,
(
x
1
,
y
2
)
,
(
x
1
,
y
3
)
,
(
x
2
,
y
2
)
∈
A
.
(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.
(
x
1
,
y
1
)
,
(
x
1
,
y
2
)
,
(
x
1
,
y
3
)
,
(
x
2
,
y
2
)
∈
A
.
Determine the largest possible number of elements in
A
A
A
.
CSMO Grade 11 Problem 8
Given the positive integer
m
≥
2
m \geq 2
m
≥
2
,
n
≥
3
n \geq 3
n
≥
3
. Define the following set
S
=
{
(
a
,
b
)
∣
a
∈
{
1
,
2
,
⋯
,
m
}
,
b
∈
{
1
,
2
,
⋯
,
n
}
}
.
S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.
S
=
{
(
a
,
b
)
∣
a
∈
{
1
,
2
,
⋯
,
m
}
,
b
∈
{
1
,
2
,
⋯
,
n
}
}
.
Let
A
A
A
be a subset of
S
S
S
. If there does not exist positive integers
x
1
,
x
2
,
x
3
,
y
1
,
y
2
,
y
3
x_1, x_2, x_3, y_1, y_2, y_3
x
1
,
x
2
,
x
3
,
y
1
,
y
2
,
y
3
such that
x
1
<
x
2
<
x
3
,
y
1
<
y
2
<
y
3
x_1 < x_2 < x_3, y_1 < y_2 < y_3
x
1
<
x
2
<
x
3
,
y
1
<
y
2
<
y
3
and
(
x
1
,
y
2
)
,
(
x
2
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
2
,
y
3
)
,
(
x
3
,
y
2
)
∈
A
.
(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.
(
x
1
,
y
2
)
,
(
x
2
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
2
,
y
3
)
,
(
x
3
,
y
2
)
∈
A
.
Determine the largest possible number of elements in
A
A
A
.
5
2
Hide problems
CSMO Grade 10 Problem 5
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in circle
O
O
O
, where
A
C
⊥
B
D
AC\perp BD
A
C
⊥
B
D
.
M
,
N
M,N
M
,
N
are the midpoint of arc
A
D
C
,
A
B
C
ADC,ABC
A
D
C
,
A
BC
.
D
O
DO
D
O
and
A
N
AN
A
N
intersect each other at
G
G
G
, the line passes through
G
G
G
and parellel to
N
C
NC
NC
intersect
C
D
CD
C
D
at
K
K
K
. Prove that
A
K
⊥
B
M
AK\perp BM
A
K
⊥
BM
.
CSMO Grade 11 Problem 5
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers,
a
≠
0
a \neq 0
a
=
0
. If the equation
2
a
x
2
+
b
x
+
c
=
0
2ax^2 + bx + c = 0
2
a
x
2
+
b
x
+
c
=
0
has real root on the interval
[
−
1
,
1
]
[-1, 1]
[
−
1
,
1
]
. Prove that
min
{
c
,
a
+
c
+
1
}
≤
max
{
∣
b
−
a
+
1
∣
,
∣
b
+
a
−
1
∣
}
,
\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},
min
{
c
,
a
+
c
+
1
}
≤
max
{
∣
b
−
a
+
1∣
,
∣
b
+
a
−
1∣
}
,
and determine the necessary and sufficient conditions of
a
,
b
,
c
a, b, c
a
,
b
,
c
for the equality case to be achieved.
4
2
Hide problems
CSMO 2017 Grade 10 Problem 4
Let
a
1
,
a
2
,
…
,
a
2017
a_1,a_2,\dots,a_{2017}
a
1
,
a
2
,
…
,
a
2017
be reals satisfied
a
1
=
a
2017
a_1=a_{2017}
a
1
=
a
2017
,
∣
a
i
+
a
i
+
2
−
2
a
i
+
1
∣
≤
1
|a_i+a_{i+2}-2a_{i+1}|\le 1
∣
a
i
+
a
i
+
2
−
2
a
i
+
1
∣
≤
1
for all
i
=
1
,
2
,
…
,
2015
i=1,2,\dots,2015
i
=
1
,
2
,
…
,
2015
. Find the maximum value of
max
1
≤
i
<
j
≤
2017
∣
a
i
−
a
j
∣
\max_{1\le i<j\le 2017}|a_i-a_j|
max
1
≤
i
<
j
≤
2017
∣
a
i
−
a
j
∣
.
CSMO Grade 11 Problem 4
For any positive integer
n
n
n
, let
D
n
D_n
D
n
denote the set of all positive divisors of
n
n
n
, and let
f
i
(
n
)
f_i(n)
f
i
(
n
)
denote the size of the set
F
i
(
n
)
=
{
a
∈
D
n
∣
a
≡
i
(
m
o
d
4
)
}
F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}
F
i
(
n
)
=
{
a
∈
D
n
∣
a
≡
i
(
mod
4
)}
where
i
=
0
,
1
,
2
,
3
i = 0, 1, 2, 3
i
=
0
,
1
,
2
,
3
. Determine the smallest positive integer
m
m
m
such that
f
0
(
m
)
+
f
1
(
m
)
−
f
2
(
m
)
−
f
3
(
m
)
=
2017
f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017
f
0
(
m
)
+
f
1
(
m
)
−
f
2
(
m
)
−
f
3
(
m
)
=
2017
.
6
2
Hide problems
CSMO Grade 10 Problem 6
The sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
=
1
2
a_1 = \frac{1}{2}
a
1
=
2
1
,
a
2
=
3
8
a_2 = \frac{3}{8}
a
2
=
8
3
, and
a
n
+
1
2
+
3
a
n
a
n
+
2
=
2
a
n
+
1
(
a
n
+
a
n
+
2
)
(
n
∈
N
∗
)
a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) (n \in \mathbb{N^*})
a
n
+
1
2
+
3
a
n
a
n
+
2
=
2
a
n
+
1
(
a
n
+
a
n
+
2
)
(
n
∈
N
∗
)
.
(
1
)
(1)
(
1
)
Determine the general formula of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
;
(
2
)
(2)
(
2
)
Prove that for any positive integer
n
n
n
, there is
0
<
a
n
<
1
2
n
+
1
0 < a_n < \frac{1}{\sqrt{2n + 1}}
0
<
a
n
<
2
n
+
1
1
.
CSMO Grade 11 Problem 6
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in circle
O
O
O
, where
A
C
⊥
B
D
AC\perp BD
A
C
⊥
B
D
.
M
M
M
be the midpoint of arc
A
D
C
ADC
A
D
C
. Circle
(
D
O
M
)
(DOM)
(
D
OM
)
intersect
D
A
,
D
C
DA,DC
D
A
,
D
C
at
E
,
F
E,F
E
,
F
. Prove that
B
E
=
B
F
BE=BF
BE
=
BF
.
1
1
Hide problems
CSMO 2017 Grade 10 Problem 1
Let
x
i
∈
{
0
,
1
}
(
i
=
1
,
2
,
⋯
,
n
)
x_i \in \{0, 1\} (i = 1, 2, \cdots, n)
x
i
∈
{
0
,
1
}
(
i
=
1
,
2
,
⋯
,
n
)
. If the function
f
=
f
(
x
1
,
x
2
,
⋯
,
x
n
)
f = f(x_1, x_2, \cdots, x_n)
f
=
f
(
x
1
,
x
2
,
⋯
,
x
n
)
only equals
0
0
0
or
1
1
1
, then define
f
f
f
as an "
n
n
n
-variable Boolean function" and denote
D
n
(
f
)
=
{
(
x
1
,
x
2
,
⋯
,
x
n
)
∣
f
(
x
1
,
x
2
,
⋯
,
x
n
)
=
0
}
D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}
D
n
(
f
)
=
{(
x
1
,
x
2
,
⋯
,
x
n
)
∣
f
(
x
1
,
x
2
,
⋯
,
x
n
)
=
0
}
.
(
1
)
(1)
(
1
)
Determine the number of
n
n
n
-variable Boolean functions;
(
2
)
(2)
(
2
)
Let
g
g
g
be a
10
10
10
-variable Boolean function satisfying
g
(
x
1
,
x
2
,
⋯
,
x
10
)
≡
1
+
x
1
+
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
1
x
2
⋯
x
10
(
m
o
d
2
)
g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}
g
(
x
1
,
x
2
,
⋯
,
x
10
)
≡
1
+
x
1
+
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
1
x
2
⋯
x
10
(
mod
2
)
Evaluate the size of the set
D
10
(
g
)
D_{10} (g)
D
10
(
g
)
and
∑
(
x
1
,
x
2
,
⋯
,
x
10
)
∈
D
10
(
g
)
(
x
1
+
x
2
+
x
3
+
⋯
+
x
10
)
\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})
(
x
1
,
x
2
,
⋯
,
x
10
)
∈
D
10
(
g
)
∑
(
x
1
+
x
2
+
x
3
+
⋯
+
x
10
)
.
3
2
Hide problems
CSMO 2017 Grade 10 Problem 3
For any positive integer
n
n
n
, let
D
n
D_n
D
n
denote the set of all positive divisors of
n
n
n
, and let
f
i
(
n
)
f_i(n)
f
i
(
n
)
denote the size of the set
F
i
(
n
)
=
{
a
∈
D
n
∣
a
≡
i
(
m
o
d
4
)
}
F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}
F
i
(
n
)
=
{
a
∈
D
n
∣
a
≡
i
(
mod
4
)}
where
i
=
1
,
2
i = 1, 2
i
=
1
,
2
. Determine the smallest positive integer
m
m
m
such that
2
f
1
(
m
)
−
f
2
(
m
)
=
2017
2f_1(m) - f_2(m) = 2017
2
f
1
(
m
)
−
f
2
(
m
)
=
2017
.
China South East Mathematical Olympiad 2017 Grade11 Q3
Let
a
1
,
a
2
,
⋯
,
a
n
+
1
>
0
a_1,a_2,\cdots,a_{n+1}>0
a
1
,
a
2
,
⋯
,
a
n
+
1
>
0
. Prove that
∑
i
−
1
n
a
i
∑
i
=
1
n
a
i
+
1
≥
∑
i
=
1
n
a
i
a
i
+
1
a
i
+
a
i
+
1
⋅
∑
i
=
1
n
(
a
i
+
a
i
+
1
)
\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})
i
−
1
∑
n
a
i
i
=
1
∑
n
a
i
+
1
≥
i
=
1
∑
n
a
i
+
a
i
+
1
a
i
a
i
+
1
⋅
i
=
1
∑
n
(
a
i
+
a
i
+
1
)
2
2
Hide problems
CSMO 2017 Grade 10 Problem 2
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. In
A
B
C
ABC
A
BC
,
A
B
≠
A
B
AB \neq AB
A
B
=
A
B
,
K
K
K
is the midpoint of the the median
A
D
AD
A
D
,
D
E
⊥
A
B
DE \perp AB
D
E
⊥
A
B
at
E
E
E
,
D
F
⊥
A
C
DF \perp AC
D
F
⊥
A
C
at
F
F
F
. The lines
K
E
KE
K
E
,
K
F
KF
K
F
intersect the line
B
C
BC
BC
at
M
M
M
,
N
N
N
, respectively. The circumcenters of
△
D
E
M
\triangle DEM
△
D
EM
,
△
D
F
N
\triangle DFN
△
D
FN
are
O
1
,
O
2
O_1, O_2
O
1
,
O
2
, respectively. Prove that
O
1
O
2
∥
B
C
O_1 O_2 \parallel BC
O
1
O
2
∥
BC
.
Who is Mr Boole?
Let
x
i
∈
{
0
,
1
}
(
i
=
1
,
2
,
⋯
,
n
)
x_i \in \{0,1\}(i=1,2,\cdots ,n)
x
i
∈
{
0
,
1
}
(
i
=
1
,
2
,
⋯
,
n
)
,if the value of function
f
=
f
(
x
1
,
x
2
,
⋯
,
x
n
)
f=f(x_1,x_2, \cdots ,x_n)
f
=
f
(
x
1
,
x
2
,
⋯
,
x
n
)
can only be
0
0
0
or
1
1
1
,then we call
f
f
f
a
n
n
n
-var Boole function,and we denote
D
n
(
f
)
=
{
(
x
1
,
x
2
,
⋯
,
x
n
)
∣
f
(
x
1
,
x
2
,
⋯
,
x
n
)
=
0
}
.
D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.
D
n
(
f
)
=
{(
x
1
,
x
2
,
⋯
,
x
n
)
∣
f
(
x
1
,
x
2
,
⋯
,
x
n
)
=
0
}
.
(
1
)
(1)
(
1
)
Find the number of
n
n
n
-var Boole function;
(
2
)
(2)
(
2
)
Let
g
g
g
be a
n
n
n
-var Boole function such that
g
(
x
1
,
x
2
,
⋯
,
x
n
)
≡
1
+
x
1
+
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
1
x
2
⋯
x
n
(
m
o
d
2
)
g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2
g
(
x
1
,
x
2
,
⋯
,
x
n
)
≡
1
+
x
1
+
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
1
x
2
⋯
x
n
(
mod
2
)
, Find the number of elements of the set
D
n
(
g
)
D_n(g)
D
n
(
g
)
,and find the maximum of
n
∈
N
+
n \in \mathbb{N}_+
n
∈
N
+
such that
∑
(
x
1
,
x
2
,
⋯
,
x
n
)
∈
D
n
(
g
)
(
x
1
+
x
2
+
⋯
+
x
n
)
≤
2017.
\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.
∑
(
x
1
,
x
2
,
⋯
,
x
n
)
∈
D
n
(
g
)
(
x
1
+
x
2
+
⋯
+
x
n
)
≤
2017.