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National and Regional Contests
China Contests
China Northern MO
2017 China Northern MO
2017 China Northern MO
Part of
China Northern MO
Subcontests
(8)
8
2
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China Northern Mathematical Olympiad 2017, Problem 8
Let
n
>
1
n>1
n
>
1
be an integer, and let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
be real numbers satisfying
x
1
,
x
2
,
.
.
.
,
x
n
∈
[
0
,
n
]
x_1, x_2, ..., x_n \in [0,n]
x
1
,
x
2
,
...
,
x
n
∈
[
0
,
n
]
with
x
1
x
2
.
.
.
x
n
=
(
n
−
x
1
)
(
n
−
x
2
)
.
.
.
(
n
−
x
n
)
x_1x_2...x_n = (n-x_1)(n-x_2)...(n-x_n)
x
1
x
2
...
x
n
=
(
n
−
x
1
)
(
n
−
x
2
)
...
(
n
−
x
n
)
. Find the maximum value of
y
=
x
1
+
x
2
+
.
.
.
+
x
n
y = x_1 + x_2 + ... + x_n
y
=
x
1
+
x
2
+
...
+
x
n
.
2017 CNMO Grade 11 P8
On Qingqing Grassland, there are 7 sheep numberd
1
,
2
,
3
,
4
,
5
,
6
,
7
1,2,3,4,5,6,7
1
,
2
,
3
,
4
,
5
,
6
,
7
and 2017 wolves numberd
1
,
2
,
⋯
,
2017
1,2,\cdots,2017
1
,
2
,
⋯
,
2017
. We have such strange rules: (1) Define
P
(
n
)
P(n)
P
(
n
)
: the number of prime numbers that are smaller than
n
n
n
. Only when
P
(
i
)
≡
j
(
m
o
d
7
)
P(i)\equiv j\pmod7
P
(
i
)
≡
j
(
mod
7
)
, wolf
i
i
i
may eat sheep
j
j
j
(he can also choose not to eat the sheep). (2) If wolf
i
i
i
eat sheep
j
j
j
, he will immediately turn into sheep
j
j
j
. (3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep. Assume that all wolves are very smart, then how many wolves will remain in the end?
7
1
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China Northern Mathematical Olympiad, Problem 7
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of the digits of the base-10 representation of an natural number
n
n
n
. For example.
S
(
2017
)
=
2
+
0
+
1
+
7
=
10
S(2017) = 2+0+1+7 = 10
S
(
2017
)
=
2
+
0
+
1
+
7
=
10
. Prove that for all primes
p
p
p
, there exists infinitely many
n
n
n
which satisfy
S
(
n
)
≡
n
m
o
d
p
S(n) \equiv n \mod p
S
(
n
)
≡
n
mod
p
.
6
2
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China Northern Mathematical Olympiad, Problem 6
Find all integers
n
n
n
such that there exists a concave pentagon which can be dissected into
n
n
n
congruent triangles.
2017 CNMO Grade 11 P6
Define
S
r
(
n
)
S_r(n)
S
r
(
n
)
: digit sum of
n
n
n
in base
r
r
r
. For example,
38
=
(
1102
)
3
,
S
3
(
38
)
=
1
+
1
+
0
+
2
=
4
38=(1102)_3,S_3(38)=1+1+0+2=4
38
=
(
1102
)
3
,
S
3
(
38
)
=
1
+
1
+
0
+
2
=
4
. Prove: (a) For any
r
>
2
r>2
r
>
2
, there exists prime
p
p
p
, for any positive intenger
n
n
n
,
S
r
(
n
)
≡
n
m
o
d
p
S_{r}(n)\equiv n\mod p
S
r
(
n
)
≡
n
mod
p
. (b) For any
r
>
1
r>1
r
>
1
and prime
p
p
p
, there exists infinitely many
n
n
n
,
S
r
(
n
)
≡
n
m
o
d
p
S_{r}(n)\equiv n\mod p
S
r
(
n
)
≡
n
mod
p
.
5
2
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China Northern Mathematical Olympiad 2017, Problem 5
Triangle
A
B
C
ABC
A
BC
has
A
B
>
A
C
AB > AC
A
B
>
A
C
and
∠
A
=
6
0
∘
\angle A = 60^\circ
∠
A
=
6
0
∘
. Let
M
M
M
be the midpoint of
B
C
BC
BC
,
N
N
N
be the point on segment
A
B
AB
A
B
such that
∠
B
N
M
=
3
0
∘
\angle BNM = 30^\circ
∠
BNM
=
3
0
∘
. Let
D
,
E
D,E
D
,
E
be points on
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively. Let
F
,
G
,
H
F, G, H
F
,
G
,
H
be the midpoints of
B
E
,
C
D
,
D
E
BE, CD, DE
BE
,
C
D
,
D
E
respectively. Let
O
O
O
be the circumcenter of triangle
F
G
H
FGH
FG
H
. Prove that
O
O
O
lies on line
M
N
MN
MN
.
2017 CNMO Grade 11 P5
Length of sides of regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is
a
a
a
. Two moving points
M
,
N
M,N
M
,
N
moves on sides
B
C
,
D
E
BC,DE
BC
,
D
E
, satisfy that
∠
M
A
N
=
π
3
\angle MAN=\frac{\pi}{3}
∠
M
A
N
=
3
π
. Prove that
A
M
⋅
A
N
−
B
M
⋅
D
N
AM\cdot AN-BM\cdot DN
A
M
⋅
A
N
−
BM
⋅
D
N
is a definite value.
4
2
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China Northern Mathematical Olympiad 2017, Problem 4
Let
Q
Q
Q
be a set of permutations of
1
,
2
,
.
.
.
,
100
1,2,...,100
1
,
2
,
...
,
100
such that for all
1
≤
a
,
b
≤
100
1\leq a,b \leq 100
1
≤
a
,
b
≤
100
,
a
a
a
can be found to the left of
b
b
b
and adjacent to
b
b
b
in at most one permutation in
Q
Q
Q
. Find the largest possible number of elements in
Q
Q
Q
.
2017 CNMO Grade 11 P4
Positive intenger
n
≥
3
n\geq3
n
≥
3
.
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
are
n
n
n
positive intengers that are pairwise coprime, satisfying that there exists
k
1
,
k
2
,
⋯
,
k
n
∈
{
−
1
,
1
}
,
∑
i
=
1
n
k
i
a
i
=
0
k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0
k
1
,
k
2
,
⋯
,
k
n
∈
{
−
1
,
1
}
,
∑
i
=
1
n
k
i
a
i
=
0
. Are there positive intengers
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots,b_n
b
1
,
b
2
,
⋯
,
b
n
, for any
k
∈
Z
+
k\in\mathbb{Z}_+
k
∈
Z
+
,
b
1
+
k
a
1
,
b
2
+
k
a
2
,
⋯
,
b
n
+
k
a
n
b_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n
b
1
+
k
a
1
,
b
2
+
k
a
2
,
⋯
,
b
n
+
k
a
n
are pairwise coprime?
3
1
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China Northern Mathematical Olympiad 2017, Problem 3
Let
D
D
D
be the midpoint of side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. Let
E
,
F
E, F
E
,
F
be points on sides
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively such that
D
E
=
D
F
DE = DF
D
E
=
D
F
. Prove that
A
E
+
A
F
=
B
E
+
C
F
⟺
∠
E
D
F
=
∠
B
A
C
AE + AF = BE + CF \iff \angle EDF = \angle BAC
A
E
+
A
F
=
BE
+
CF
⟺
∠
E
D
F
=
∠
B
A
C
.
2
1
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China Northern Mathematical Olympiad 2017, Problem 2
Prove that there exist infinitely many integers
n
n
n
which satisfy
201
7
2
∣
1
n
+
2
n
+
.
.
.
+
201
7
n
2017^2 | 1^n + 2^n + ... + 2017^n
201
7
2
∣
1
n
+
2
n
+
...
+
201
7
n
.
1
2
Hide problems
China Northern Mathematical Olympiad 2017, Problem 1
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined as follows:
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
1
3
a_2 = \frac{1}{3}
a
2
=
3
1
, and for all
n
≥
1
,
n \geq 1,
n
≥
1
,
(
1
+
a
n
)
(
1
+
a
n
+
2
)
(
1
+
a
n
+
1
)
2
=
a
n
a
n
+
2
a
n
+
1
2
\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}
(
1
+
a
n
+
1
)
2
(
1
+
a
n
)
(
1
+
a
n
+
2
)
=
a
n
+
1
2
a
n
a
n
+
2
. Prove that, for all
n
≥
1
n \geq 1
n
≥
1
,
a
1
+
a
2
+
.
.
.
+
a
n
<
34
21
a_1 + a_2 + ... + a_n < \frac{34}{21}
a
1
+
a
2
+
...
+
a
n
<
21
34
.
2017 CNMO Grade 11 P1
Define sequence
(
a
n
)
:
a
1
=
e
,
a
2
=
e
3
,
e
1
−
k
a
n
k
+
2
=
a
n
+
1
a
n
−
1
2
k
(a_n):a_1=\text{e},a_2=\text{e}^3,\text{e}^{1-k}a_n^{k+2}=a_{n+1}a_{n-1}^{2k}
(
a
n
)
:
a
1
=
e
,
a
2
=
e
3
,
e
1
−
k
a
n
k
+
2
=
a
n
+
1
a
n
−
1
2
k
for all
n
≥
2
n\geq2
n
≥
2
, where
k
k
k
is a positive real number. Find
∏
i
=
1
2017
a
i
\prod_{i=1}^{2017}a_i
∏
i
=
1
2017
a
i
.