MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2009 China Second Round Olympiad
2009 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
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3 x 9 matrix with conditions on the first three columns
Let
P
=
[
a
i
j
]
3
×
9
P=[a_{ij}]_{3\times 9}
P
=
[
a
ij
]
3
×
9
be a
3
×
9
3\times 9
3
×
9
matrix where
a
i
j
≥
0
a_{ij}\ge 0
a
ij
≥
0
for all
i
,
j
i,j
i
,
j
. The following conditions are given: [*]Every row consists of distinct numbers; [*]
∑
i
=
1
3
x
i
j
=
1
\sum_{i=1}^{3}x_{ij}=1
∑
i
=
1
3
x
ij
=
1
for
1
≤
j
≤
6
1\le j\le 6
1
≤
j
≤
6
; [*]
x
17
=
x
28
=
x
39
=
0
x_{17}=x_{28}=x_{39}=0
x
17
=
x
28
=
x
39
=
0
; [*]
x
i
j
>
1
x_{ij}>1
x
ij
>
1
for all
1
≤
i
≤
3
1\le i\le 3
1
≤
i
≤
3
and
7
≤
j
≤
9
7\le j\le 9
7
≤
j
≤
9
such that
j
−
i
≠
6
j-i\not= 6
j
−
i
=
6
. [*]The first three columns of
P
P
P
satisfy the following property
(
R
)
(R)
(
R
)
: for an arbitrary column
[
x
1
k
,
x
2
k
,
x
3
k
]
T
[x_{1k},x_{2k},x_{3k}]^T
[
x
1
k
,
x
2
k
,
x
3
k
]
T
,
1
≤
k
≤
9
1\le k\le 9
1
≤
k
≤
9
, there exists an
i
∈
{
1
,
2
,
3
}
i\in\{1,2,3\}
i
∈
{
1
,
2
,
3
}
such that
x
i
k
≤
u
i
=
min
(
x
i
1
,
x
i
2
,
x
i
3
)
x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})
x
ik
≤
u
i
=
min
(
x
i
1
,
x
i
2
,
x
i
3
)
. Prove that: a) the elements
u
1
,
u
2
,
u
3
u_1,u_2,u_3
u
1
,
u
2
,
u
3
come from three different columns; b) if a column
[
x
1
l
,
x
2
l
,
x
3
l
]
T
[x_{1l},x_{2l},x_{3l}]^T
[
x
1
l
,
x
2
l
,
x
3
l
]
T
of
P
P
P
, where
l
≥
4
l\ge 4
l
≥
4
, satisfies the condition that after replacing the third column of
P
P
P
by it, the first three columns of the newly obtained matrix
P
′
P'
P
′
still have property
(
R
)
(R)
(
R
)
, then this column uniquely exists.
3
1
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Gcd of binomial coefficient and integer is 1
Let
k
,
l
k,l
k
,
l
be two given integers. Prove that there exist infinite many integers
m
≥
k
m\ge k
m
≥
k
such that
gcd
(
(
m
k
)
,
l
)
=
1
\gcd\left(\binom{m}{k},l\right)=1
g
cd
(
(
k
m
)
,
l
)
=
1
.
2
1
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Upper and lower bound of sum and ln(n)
Let
n
n
n
be a positive integer. Prove that
−
1
<
∑
k
=
1
n
k
k
2
+
1
−
ln
n
≤
1
2
-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}
−
1
<
k
=
1
∑
n
k
2
+
1
k
−
ln
n
≤
2
1
1
1
Hide problems
The points Q,I,J,T are concylic
Let
ω
\omega
ω
be the circumcircle of acute triangle
A
B
C
ABC
A
BC
where
∠
A
<
∠
B
\angle A<\angle B
∠
A
<
∠
B
and
M
,
N
M,N
M
,
N
be the midpoints of minor arcs
B
C
,
A
C
BC,AC
BC
,
A
C
of
ω
\omega
ω
respectively. The line
P
C
PC
PC
is parallel to
M
N
MN
MN
, intersecting
ω
\omega
ω
at
P
P
P
(different from
C
C
C
). Let
I
I
I
be the incentre of
A
B
C
ABC
A
BC
and let
P
I
PI
P
I
intersect
ω
\omega
ω
again at the point
T
T
T
. 1) Prove that
M
P
⋅
M
T
=
N
P
⋅
N
T
MP\cdot MT=NP\cdot NT
MP
⋅
MT
=
NP
⋅
NT
; 2) Let
Q
Q
Q
be an arbitrary point on minor arc
A
B
AB
A
B
and
I
,
J
I,J
I
,
J
be the incentres of triangles
A
Q
C
,
B
C
Q
AQC,BCQ
A
QC
,
BCQ
. Prove that
Q
,
I
,
J
,
T
Q,I,J,T
Q
,
I
,
J
,
T
are concyclic.