MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2012 China National Olympiad
2012 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
There exist a increasing sequence
Prove for any
M
>
2
M>2
M
>
2
, there exists an increasing sequence of positive integers
a
1
<
a
2
<
…
a_1<a_2<\ldots
a
1
<
a
2
<
…
satisfying: 1)
a
i
>
M
i
a_i>M^i
a
i
>
M
i
for any
i
i
i
; 2) There exists a positive integer
m
m
m
and
b
1
,
b
2
,
…
,
b
m
∈
{
−
1
,
1
}
b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}
b
1
,
b
2
,
…
,
b
m
∈
{
−
1
,
1
}
, satisfying
n
=
a
1
b
1
+
a
2
b
2
+
…
+
a
m
b
m
n=a_1b_1+a_2b_2+\ldots +a_mb_m
n
=
a
1
b
1
+
a
2
b
2
+
…
+
a
m
b
m
if and only if
n
∈
Z
/
{
0
}
n\in\mathbb{Z}/ \{0\}
n
∈
Z
/
{
0
}
.
Least k for which there exist x,y,z
Find the smallest positive integer
k
k
k
such that, for any subset
A
A
A
of
S
=
{
1
,
2
,
…
,
2012
}
S=\{1,2,\ldots,2012\}
S
=
{
1
,
2
,
…
,
2012
}
with
∣
A
∣
=
k
|A|=k
∣
A
∣
=
k
, there exist three elements
x
,
y
,
z
x,y,z
x
,
y
,
z
in
A
A
A
such that
x
=
a
+
b
x=a+b
x
=
a
+
b
,
y
=
b
+
c
y=b+c
y
=
b
+
c
,
z
=
c
+
a
z=c+a
z
=
c
+
a
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are in
S
S
S
and are distinct integers.Proposed by Huawei Zhu
2
2
Hide problems
ab+bc+ca
Consider a square-free even integer
n
n
n
and a prime
p
p
p
, such that 1)
(
n
,
p
)
=
1
(n,p)=1
(
n
,
p
)
=
1
; 2)
p
≤
2
n
p\le 2\sqrt{n}
p
≤
2
n
; 3) There exists an integer
k
k
k
such that
p
∣
n
+
k
2
p|n+k^2
p
∣
n
+
k
2
. Prove that there exists pairwise distinct positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
n
=
a
b
+
b
c
+
c
a
n=ab+bc+ca
n
=
ab
+
b
c
+
c
a
.Proposed by Hongbing Yu
Matrices that can be converted to 0 by certain operations
Let
p
p
p
be a prime. We arrange the numbers in
{
1
,
2
,
…
,
p
2
}
{\{1,2,\ldots ,p^2} \}
{
1
,
2
,
…
,
p
2
}
as a
p
×
p
p \times p
p
×
p
matrix
A
=
(
a
i
j
)
A = ( a_{ij} )
A
=
(
a
ij
)
. Next we can select any row or column and add
1
1
1
to every number in it, or subtract
1
1
1
from every number in it. We call the arrangement good if we can change every number of the matrix to
0
0
0
in a finite number of such moves. How many good arrangements are there?
1
2
Hide problems
AP bisects angle BAC
In the triangle
A
B
C
ABC
A
BC
,
∠
A
\angle A
∠
A
is biggest. On the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
, let
D
D
D
be the midpoint of
A
B
C
^
\widehat{ABC}
A
BC
and
E
E
E
be the midpoint of
A
C
B
^
\widehat{ACB}
A
CB
. The circle
c
1
c_1
c
1
passes through
A
,
B
A,B
A
,
B
and is tangent to
A
C
AC
A
C
at
A
A
A
, the circle
c
2
c_2
c
2
passes through
A
,
E
A,E
A
,
E
and is tangent
A
D
AD
A
D
at
A
A
A
.
c
1
c_1
c
1
and
c
2
c_2
c
2
intersect at
A
A
A
and
P
P
P
. Prove that
A
P
AP
A
P
bisects
∠
B
A
C
\angle BAC
∠
B
A
C
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A
A
A
", (0.81,3.72), NE * labelscalefactor); label("
c
1
c_1
c
1
", (-2.81,3.53), NE * labelscalefactor); dot((3.56,1.53),dotstyle); label("
c
2
c_2
c
2
", (3.43,3.98), NE * labelscalefactor); dot((1.05,-0.43),dotstyle); label("
P
P
P
", (0.5,-0.43), NE * labelscalefactor); dot((-2.72,-1.27),dotstyle); label("
B
B
B
", (-3.02,-1.57), NE * labelscalefactor); dot((2.71,-1.54),dotstyle); label("
E
E
E
", (2.71,-1.86), NE * labelscalefactor); dot((3.13,-0.65),dotstyle); label("
C
C
C
", (3.39,-0.9), NE * labelscalefactor); dot((-2.41,-1.74),dotstyle); label("
D
D
D
", (-2.78,-2.07), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]
Find the maximum of F
Let
f
(
x
)
=
(
x
+
a
)
(
x
+
b
)
f(x)=(x + a)(x + b)
f
(
x
)
=
(
x
+
a
)
(
x
+
b
)
where
a
,
b
>
0
a,b>0
a
,
b
>
0
. For any reals
x
1
,
x
2
,
…
,
x
n
⩾
0
x_1,x_2,\ldots ,x_n\geqslant 0
x
1
,
x
2
,
…
,
x
n
⩾
0
satisfying
x
1
+
x
2
+
…
+
x
n
=
1
x_1+x_2+\ldots +x_n =1
x
1
+
x
2
+
…
+
x
n
=
1
, find the maximum of
F
=
∑
1
⩽
i
<
j
⩽
n
min
{
f
(
x
i
)
,
f
(
x
j
)
}
F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}}
F
=
1
⩽
i
<
j
⩽
n
∑
min
{
f
(
x
i
)
,
f
(
x
j
)
}
.