MathDB
Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2012 China Western Mathematical Olympiad
2012 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(4)
3
2
Hide problems
Disjoint or entirely contained subsets
Let
A
A
A
be a set of
n
n
n
elements and
A
1
,
A
2
,
.
.
.
A
k
A_1, A_2, ... A_k
A
1
,
A
2
,
...
A
k
subsets of
A
A
A
such that for any
2
2
2
distinct subsets
A
i
,
A
j
A_i, A_j
A
i
,
A
j
either they are disjoint or one contains the other. Find the maximum value of
k
k
k
Operations on a grid
Let
n
n
n
be a positive integer
≥
2
\geq 2
≥
2
. Consider a
n
n
n
by
n
n
n
grid with all entries
1
1
1
. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all
n
n
n
such that it is possible after a finite sequence of operations to reach a
n
n
n
by
n
n
n
grid with all entries
−
1
-1
−
1
4
2
Hide problems
Geometric inequality
P
P
P
is a point in the
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
ω
\omega
ω
is the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
.
B
P
∩
ω
=
{
B
,
B
1
}
BP \cap \omega = \left\{ {B,{B_1}} \right\}
BP
∩
ω
=
{
B
,
B
1
}
,
C
P
∩
ω
=
{
C
,
C
1
}
CP \cap \omega = \left\{ {C,{C_1}} \right\}
CP
∩
ω
=
{
C
,
C
1
}
,
P
E
⊥
A
C
PE \bot AC
PE
⊥
A
C
，
P
F
⊥
A
B
PF \bot AB
PF
⊥
A
B
. The radius of the inscribed circle and circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
is
r
,
R
r,R
r
,
R
. Prove
E
F
B
1
C
1
⩾
r
R
\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}
B
1
C
1
EF
⩾
R
r
.
Find prime number
Find all prime number
p
p
p
, such that there exist an infinite number of positive integer
n
n
n
satisfying the following condition:
p
∣
n
n
+
1
+
(
n
+
1
)
n
.
p|n^{ n+1}+(n+1)^n.
p
∣
n
n
+
1
+
(
n
+
1
)
n
.
(September 29, 2012, Hohhot)
2
2
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Isosceles triangles in regular polygon
Show that among any
n
≥
3
n\geq 3
n
≥
3
vertices of a regular
(
2
n
−
1
)
(2n-1)
(
2
n
−
1
)
-gon we can find
3
3
3
of them forming an isosceles triangle.
Find integer
Define a sequence
{
a
n
}
\{a_n\}
{
a
n
}
by
a
0
=
1
2
,
a
n
+
1
=
a
n
+
a
n
2
2012
,
(
n
=
0
,
1
,
2
,
⋯
)
,
a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),
a
0
=
2
1
,
a
n
+
1
=
a
n
+
2012
a
n
2
,
(
n
=
0
,
1
,
2
,
⋯
)
,
find integer
k
k
k
such that
a
k
<
1
<
a
k
+
1
.
a_{k}<1<a_{k+1}.
a
k
<
1
<
a
k
+
1
.
(September 29, 2012, Hohhot)
1
2
Hide problems
Find smallest positive integer
Find the smallest positive integer
m
m
m
satisfying the following condition: for all prime numbers
p
p
p
such that
p
>
3
p>3
p
>
3
,have
105
∣
9
p
2
−
2
9
p
+
m
.
105|9^{ p^2}-29^p+m.
105∣
9
p
2
−
2
9
p
+
m
.
(September 28, 2012, Hohhot)
through the midpoint of $OH$
O
O
O
is the circumcenter of acute
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
H
H
H
is the Orthocenter.
A
D
⊥
B
C
AD \bot BC
A
D
⊥
BC
,
E
F
EF
EF
is the perpendicular bisector of
A
O
AO
A
O
,
D
,
E
D,E
D
,
E
on the
B
C
BC
BC
. Prove that the circumcircle of
Δ
A
D
E
\Delta ADE
Δ
A
D
E
through the midpoint of
O
H
OH
O
H
.