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Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2016 China Northern MO
2016 China Northern MO
Part of
China Northern MO
Subcontests
(8)
8
2
Hide problems
2016 CNMO Grade 11 P8
Given a set
I
=
{
(
x
1
,
x
2
,
x
3
,
x
4
)
∣
x
i
∈
{
1
,
2
,
⋯
,
11
}
}
I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}
I
=
{(
x
1
,
x
2
,
x
3
,
x
4
)
∣
x
i
∈
{
1
,
2
,
⋯
,
11
}}
.
A
⊆
I
A\subseteq I
A
⊆
I
, satisfying that for any
(
x
1
,
x
2
,
x
3
,
x
4
)
,
(
y
1
,
y
2
,
y
3
,
y
4
)
∈
A
(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A
(
x
1
,
x
2
,
x
3
,
x
4
)
,
(
y
1
,
y
2
,
y
3
,
y
4
)
∈
A
, there exists
i
,
j
(
1
≤
i
<
j
≤
4
)
i,j(1\leq i<j\leq4)
i
,
j
(
1
≤
i
<
j
≤
4
)
,
(
x
i
−
x
j
)
(
y
i
−
y
j
)
<
0
(x_i-x_j)(y_i-y_j)<0
(
x
i
−
x
j
)
(
y
i
−
y
j
)
<
0
. Find the maximum value of
∣
A
∣
|A|
∣
A
∣
.
2016 CNMO Grade 10 P8
Set
A
=
{
1
,
2
,
⋯
,
n
}
A=\{1,2,\cdots,n\}
A
=
{
1
,
2
,
⋯
,
n
}
. If there exists nonempty sets
B
,
C
B,C
B
,
C
, such that
B
∩
C
=
∅
,
B
∪
C
=
A
B\cap C=\varnothing,B\cup C=A
B
∩
C
=
∅
,
B
∪
C
=
A
. Sum of Squares of all elements in
B
B
B
is
M
M
M
, Sum of Squares of all elements in
C
C
C
is
N
N
N
,
M
−
N
=
2016
M-N=2016
M
−
N
=
2016
. Find the minimum value of
n
n
n
.
7
1
Hide problems
2016 CNMO Grade 10/11 P7
Define sequence
(
a
n
)
:
a
n
=
2
n
+
3
n
+
6
n
+
1
(
n
∈
Z
+
)
(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)
(
a
n
)
:
a
n
=
2
n
+
3
n
+
6
n
+
1
(
n
∈
Z
+
)
. Are there intenger
k
≥
2
k\geq2
k
≥
2
, satisfying that
gcd
(
k
,
a
i
)
=
1
\gcd(k,a_i)=1
g
cd
(
k
,
a
i
)
=
1
for all
k
∈
Z
+
k\in\mathbb{Z}_+
k
∈
Z
+
? If yes, find the smallest
k
k
k
. If not, prove this.
6
1
Hide problems
2016 CNMO Grade 10/11 P6
Four points
B
,
E
,
A
,
F
B,E,A,F
B
,
E
,
A
,
F
lie on line
A
B
AB
A
B
in order, four points
C
,
G
,
D
,
H
C,G,D,H
C
,
G
,
D
,
H
lie on line
C
D
CD
C
D
in order, satisfying:
A
E
E
B
=
A
F
F
B
=
D
G
G
C
=
D
H
H
C
=
A
D
B
C
.
\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.
EB
A
E
=
FB
A
F
=
GC
D
G
=
H
C
DH
=
BC
A
D
.
Prove that
F
H
⊥
E
G
FH\perp EG
F
H
⊥
EG
.
5
2
Hide problems
2016 CNMO Grade P5
Let
θ
i
∈
(
0
,
π
2
)
(
i
=
1
,
2
,
⋯
,
n
)
\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)
θ
i
∈
(
0
,
2
π
)
(
i
=
1
,
2
,
⋯
,
n
)
. Prove:
(
∑
i
=
1
n
tan
θ
i
)
(
∑
i
=
1
n
cot
θ
i
)
≥
(
∑
i
=
1
n
sin
θ
i
)
2
+
(
∑
i
=
1
n
cos
θ
i
)
2
.
(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.
(
i
=
1
∑
n
tan
θ
i
)
(
i
=
1
∑
n
cot
θ
i
)
≥
(
i
=
1
∑
n
sin
θ
i
)
2
+
(
i
=
1
∑
n
cos
θ
i
)
2
.
2016 CNMO Grade 11 P5
a
1
=
2
,
a
n
+
1
=
2
n
+
1
a
n
(
n
+
1
2
)
a
n
+
2
n
(
n
∈
Z
+
)
a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)
a
1
=
2
,
a
n
+
1
=
(
n
+
2
1
)
a
n
+
2
n
2
n
+
1
a
n
(
n
∈
Z
+
)
(a) Find
a
n
a_n
a
n
. (b) Let
b
n
=
n
3
+
2
n
2
+
2
n
+
2
n
(
n
+
1
)
(
n
2
+
1
)
a
n
b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}
b
n
=
n
(
n
+
1
)
(
n
2
+
1
)
a
n
n
3
+
2
n
2
+
2
n
+
2
. Find
S
n
=
∑
i
=
1
n
b
i
S_n=\sum_{i=1}^nb_i
S
n
=
∑
i
=
1
n
b
i
.
4
1
Hide problems
2016 CNMO Grade 10/11 P4
Can we put intengers
1
,
2
,
⋯
,
12
1,2,\cdots,12
1
,
2
,
⋯
,
12
on a circle, number them
a
1
,
a
2
,
⋯
,
a
12
a_1,a_2,\cdots,a_{12}
a
1
,
a
2
,
⋯
,
a
12
in order. For any
1
≤
i
<
j
≤
12
1\leq i<j\leq12
1
≤
i
<
j
≤
12
,
∣
a
i
−
a
j
∣
≠
∣
i
−
j
∣
|a_i-a_j|\neq|i-j|
∣
a
i
−
a
j
∣
=
∣
i
−
j
∣
?
3
2
Hide problems
2016 CNMO Grade 10 P3
Prove: (a) There are infinitely many positive intengers
n
n
n
, satisfying:
gcd
(
n
,
[
2
n
]
)
=
1.
\gcd(n,[\sqrt2n])=1.
g
cd
(
n
,
[
2
n
])
=
1.
(b) There are infinitely many positive intengers
n
n
n
, satisfying:
gcd
(
n
,
[
2
n
]
)
>
1.
\gcd(n,[\sqrt2n])>1.
g
cd
(
n
,
[
2
n
])
>
1.
2016 CNMO Grade 11 P3
m
(
m
>
1
)
m(m>1)
m
(
m
>
1
)
is an intenger, define
(
a
n
)
(a_n)
(
a
n
)
:
a
0
=
m
,
a
n
=
φ
(
a
n
−
1
)
a_0=m,a_{n}=\varphi(a_{n-1})
a
0
=
m
,
a
n
=
φ
(
a
n
−
1
)
for all positive intenger
n
n
n
. If for all nonnegative intenger
k
k
k
,
a
k
+
1
∣
a
k
a_{k+1}\mid a_k
a
k
+
1
∣
a
k
, find all
m
m
m
that is not larger than
2016
2016
2016
. Note:
φ
(
n
)
\varphi(n)
φ
(
n
)
means Euler Function.
2
2
Hide problems
2016 CNMO Grade 10 P2
In isosceles triangle
A
B
C
ABC
A
BC
,
∠
C
A
B
=
∠
C
B
A
=
α
\angle CAB=\angle CBA=\alpha
∠
C
A
B
=
∠
CB
A
=
α
, points
P
,
Q
P,Q
P
,
Q
are on different sides of line
A
B
AB
A
B
, and
∠
C
A
P
=
∠
A
B
Q
=
β
,
∠
C
B
P
=
∠
B
A
Q
=
γ
\angle CAP=\angle ABQ=\beta,\angle CBP=\angle BAQ=\gamma
∠
C
A
P
=
∠
A
BQ
=
β
,
∠
CBP
=
∠
B
A
Q
=
γ
. Prove that
P
,
C
,
Q
P,C,Q
P
,
C
,
Q
are colinear.
2016 CNMO Grade 11 P2
Inscribed Triangle
A
B
C
ABC
A
BC
on circle
⊙
O
\odot O
⊙
O
. Bisector of
∠
A
B
C
\angle ABC
∠
A
BC
intersects
⊙
O
\odot O
⊙
O
at
D
D
D
. Two lines
P
B
PB
PB
and
P
C
PC
PC
that are tangent to
⊙
O
\odot O
⊙
O
intersect at
P
P
P
.
P
D
PD
P
D
intersects
A
C
AC
A
C
at
E
E
E
,
⊙
O
\odot O
⊙
O
at
F
F
F
.
M
M
M
is the midpoint of
B
C
BC
BC
. Prove that
M
,
F
,
C
,
E
M,F,C,E
M
,
F
,
C
,
E
are concyclic.
1
1
Hide problems
2016 CNMO Grade 10/11 P1
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
are positive real numbers,
a
1
+
a
2
+
⋯
,
a
n
=
1
a_1+a_2+\cdots,a_n=1
a
1
+
a
2
+
⋯
,
a
n
=
1
. Prove that
∑
m
=
1
n
a
m
∏
k
=
1
m
(
1
+
a
k
)
≤
1
−
1
2
n
.
\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.
m
=
1
∑
n
k
=
1
∏
m
(
1
+
a
k
)
a
m
≤
1
−
2
n
1
.