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Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2017 China National Olympiad
2017 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
6
1
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China Mathematical Olympiad 2017 Q6
Given an integer
n
≥
2
n \geq2
n
≥
2
and real numbers
a
,
b
a,b
a
,
b
such that
0
<
a
<
b
0<a<b
0
<
a
<
b
. Let
x
1
,
x
2
,
…
,
x
n
∈
[
a
,
b
]
x_1,x_2,\ldots, x_n\in [a,b]
x
1
,
x
2
,
…
,
x
n
∈
[
a
,
b
]
be real numbers. Find the maximum value of
x
1
2
x
2
+
x
2
2
x
3
+
⋯
+
x
n
−
1
2
x
n
+
x
n
2
x
1
x
1
+
x
2
+
⋯
+
x
n
−
1
+
x
n
.
\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.
x
1
+
x
2
+
⋯
+
x
n
−
1
+
x
n
x
2
x
1
2
+
x
3
x
2
2
+
⋯
+
x
n
x
n
−
1
2
+
x
1
x
n
2
.
4
1
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Arranging permutations in a circle
Let
n
≥
2
n \geq 2
n
≥
2
be a natural number. For any two permutations of
(
1
,
2
,
⋯
,
n
)
(1,2,\cdots,n)
(
1
,
2
,
⋯
,
n
)
, say
α
=
(
a
1
,
a
2
,
⋯
,
a
n
)
\alpha = (a_1,a_2,\cdots,a_n)
α
=
(
a
1
,
a
2
,
⋯
,
a
n
)
and
β
=
(
b
1
,
b
2
,
⋯
,
b
n
)
,
\beta = (b_1,b_2,\cdots,b_n),
β
=
(
b
1
,
b
2
,
⋯
,
b
n
)
,
if there exists a natural number
k
≤
n
k \leq n
k
≤
n
such that
b
i
=
{
a
k
+
1
−
i
,
1
≤
i
≤
k
;
a
i
,
k
<
i
≤
n
,
b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}
b
i
=
{
a
k
+
1
−
i
,
a
i
,
1
≤
i
≤
k
;
k
<
i
≤
n
,
we call
α
\alpha
α
a friendly permutation of
β
\beta
β
.Prove that it is possible to enumerate all possible permutations of
(
1
,
2
,
⋯
,
n
)
(1,2,\cdots,n)
(
1
,
2
,
⋯
,
n
)
as
P
1
,
P
2
,
⋯
,
P
m
P_1,P_2,\cdots,P_m
P
1
,
P
2
,
⋯
,
P
m
such that for all
i
=
1
,
2
,
⋯
,
m
i = 1,2,\cdots,m
i
=
1
,
2
,
⋯
,
m
,
P
i
+
1
P_{i+1}
P
i
+
1
is a friendly permutation of
P
i
P_i
P
i
where
m
=
n
!
m = n!
m
=
n
!
and
P
m
+
1
=
P
1
P_{m+1} = P_1
P
m
+
1
=
P
1
.
5
1
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Partition divisors into AP and GP
Let
D
n
D_n
D
n
be the set of divisors of
n
n
n
. Find all natural
n
n
n
such that it is possible to split
D
n
D_n
D
n
into two disjoint sets
A
A
A
and
G
G
G
, both containing at least three elements each, such that the elements in
A
A
A
form an arithmetic progression while the elements in
G
G
G
form a geometric progression.
2
1
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Concyclic iff collinear
In acute triangle
A
B
C
ABC
A
BC
, let
⊙
O
\odot O
⊙
O
be its circumcircle,
⊙
I
\odot I
⊙
I
be its incircle. Tangents at
B
,
C
B,C
B
,
C
to
⊙
O
\odot O
⊙
O
meet at
L
L
L
,
⊙
I
\odot I
⊙
I
touches
B
C
BC
BC
at
D
D
D
.
A
Y
AY
A
Y
is perpendicular to
B
C
BC
BC
at
Y
Y
Y
,
A
O
AO
A
O
meets
B
C
BC
BC
at
X
X
X
, and
O
I
OI
O
I
meets
⊙
O
\odot O
⊙
O
at
P
,
Q
P,Q
P
,
Q
. Prove that
P
,
Q
,
X
,
Y
P,Q,X,Y
P
,
Q
,
X
,
Y
are concyclic if and only if
A
,
D
,
L
A,D,L
A
,
D
,
L
are collinear.
1
1
Hide problems
China Mathematical Olympiad 2017 Q1
The sequences
{
u
n
}
\{u_{n}\}
{
u
n
}
and
{
v
n
}
\{v_{n}\}
{
v
n
}
are defined by
u
0
=
u
1
=
1
u_{0} =u_{1} =1
u
0
=
u
1
=
1
,
u
n
=
2
u
n
−
1
−
3
u
n
−
2
u_{n}=2u_{n-1}-3u_{n-2}
u
n
=
2
u
n
−
1
−
3
u
n
−
2
(
n
≥
2
)
(n\geq2)
(
n
≥
2
)
,
v
0
=
a
,
v
1
=
b
,
v
2
=
c
v_{0} =a, v_{1} =b , v_{2}=c
v
0
=
a
,
v
1
=
b
,
v
2
=
c
,
v
n
=
v
n
−
1
−
3
v
n
−
2
+
27
v
n
−
3
v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}
v
n
=
v
n
−
1
−
3
v
n
−
2
+
27
v
n
−
3
(
n
≥
3
)
(n\geq3)
(
n
≥
3
)
. There exists a positive integer
N
N
N
such that when
n
>
N
n> N
n
>
N
, we have
u
n
∣
v
n
u_{n}\mid v_{n}
u
n
∣
v
n
. Prove that
3
a
=
2
b
+
c
3a=2b+c
3
a
=
2
b
+
c
.
3
1
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Basic lines in rectangular partition
Consider a rectangle
R
R
R
partitioned into
2016
2016
2016
smaller rectangles such that the sides of each smaller rectangle is parallel to one of the sides of the original rectangle. Call the corners of each rectangle a vertex. For any segment joining two vertices, call it basic if no other vertex lie on it. (The segments must be part of the partitioning.) Find the maximum/minimum possible number of basic segments over all possible partitions of
R
R
R
.