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Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2013 China Western Mathematical Olympiad
2013 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(8)
4
1
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Coins..
There are
n
n
n
coins in a row,
n
≥
2
n\geq 2
n
≥
2
. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a
m
o
v
e
move
m
o
v
e
. No move is allowed if all
n
n
n
coins are tails. Suppose
m
−
1
m-1
m
−
1
coins are heads at the initial stage, determine if there is a way to carry out
⌊
2
m
3
⌋
\lfloor\frac {2^m}{3}\rfloor
⌊
3
2
m
⌋
moves
7
1
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Find all integers n
Label sides of a regular
n
n
n
-gon in clockwise direction in order 1,2,..,n. Determine all integers n (
n
≥
4
n\geq 4
n
≥
4
) satisfying the following conditions: (1)
n
−
3
n-3
n
−
3
non-intersecting diagonals in the
n
n
n
-gon are selected, which subdivide the
n
n
n
-gon into
n
−
2
n-2
n
−
2
non-overlapping triangles; (2) each of the chosen
n
−
3
n-3
n
−
3
diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;
6
1
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tangents and angle bisector
Let
P
A
,
P
B
PA, PB
P
A
,
PB
be tangents to a circle centered at
O
O
O
, and
C
C
C
a point on the minor arc
A
B
AB
A
B
. The perpendicular from
C
C
C
to
P
C
PC
PC
intersects internal angle bisectors of
A
O
C
,
B
O
C
AOC,BOC
A
OC
,
BOC
at
D
,
E
D,E
D
,
E
. Show that
C
D
=
C
E
CD=CE
C
D
=
CE
5
1
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2013 China Western Mathematical Olympiad ,Problem 5
A nonempty set
A
A
A
is called an
n
n
n
-level-good set if
A
⊆
{
1
,
2
,
3
,
…
,
n
}
A \subseteq \{1,2,3,\ldots,n\}
A
⊆
{
1
,
2
,
3
,
…
,
n
}
and
∣
A
∣
≤
min
x
∈
A
x
|A| \le \min_{x\in A} x
∣
A
∣
≤
min
x
∈
A
x
(where
∣
A
∣
|A|
∣
A
∣
denotes the number of elements in
A
A
A
and
min
x
∈
A
x
\min_{x\in A} x
min
x
∈
A
x
denotes the minimum of the elements in
A
A
A
). Let
a
n
a_n
a
n
be the number of
n
n
n
-level-good sets. Prove that for all positive integers
n
n
n
we have
a
n
+
2
=
a
n
+
1
+
a
n
+
1
a_{n+2}=a_{n+1}+a_{n}+1
a
n
+
2
=
a
n
+
1
+
a
n
+
1
.
8
1
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2013 China Western Mathematical Olympiad ,Problem 8
Find all positive integers
a
a
a
such that for any positive integer
n
≥
5
n\ge 5
n
≥
5
we have
2
n
−
n
2
∣
a
n
−
n
a
2^n-n^2\mid a^n-n^a
2
n
−
n
2
∣
a
n
−
n
a
.
3
1
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Excenters and Extouch
Let
A
B
C
ABC
A
BC
be a triangle, and
B
1
,
C
1
B_1,C_1
B
1
,
C
1
be its excenters opposite
B
,
C
B,C
B
,
C
.
B
2
,
C
2
B_2,C_2
B
2
,
C
2
are reflections of
B
1
,
C
1
B_1,C_1
B
1
,
C
1
across midpoints of
A
C
,
A
B
AC,AB
A
C
,
A
B
. Let
D
D
D
be the extouch at
B
C
BC
BC
. Show that
A
D
AD
A
D
is perpendicular to
B
2
C
2
B_2C_2
B
2
C
2
1
1
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2013 China Western Mathematical Olympiad ,Problem 1
Does there exist any integer
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
2
b
c
+
2
,
a
b
2
c
+
2
,
a
b
c
2
+
2
a^2bc+2,ab^2c+2,abc^2+2
a
2
b
c
+
2
,
a
b
2
c
+
2
,
ab
c
2
+
2
are perfect squares?
2
1
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2013 China Western Mathematical Olympiad ,Problem 2
Let the integer
n
≥
2
n \ge 2
n
≥
2
, and the real numbers
x
1
,
x
2
,
⋯
,
x
n
∈
[
0
,
1
]
x_1,x_2,\cdots,x_n\in \left[0,1\right]
x
1
,
x
2
,
⋯
,
x
n
∈
[
0
,
1
]
.Prove that
∑
1
≤
k
<
j
≤
n
k
x
k
x
j
≤
n
−
1
3
∑
k
=
1
n
k
x
k
.
\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.
1
≤
k
<
j
≤
n
∑
k
x
k
x
j
≤
3
n
−
1
k
=
1
∑
n
k
x
k
.