MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2007 China Northern MO
2007 China Northern MO
Part of
China Northern MO
Subcontests
(4)
4
2
Hide problems
Points and Colors
For every point on the plane, one of
n
n
n
colors are colored to it such that:
(
1
)
(1)
(
1
)
Every color is used infinitely many times.
(
2
)
(2)
(
2
)
There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of
n
n
n
such that there exist four concyclic points with pairwise distinct colors.
Triangle sides.
The inradius of triangle
A
B
C
ABC
A
BC
is
1
1
1
and the side lengths of
A
B
C
ABC
A
BC
are all integers. Prove that triangle
A
B
C
ABC
A
BC
is right-angled.
3
2
Hide problems
Sequence
Sequence
{
a
n
}
\{a_{n}\}
{
a
n
}
is defined by
a
1
=
2007
,
a
n
+
1
=
a
n
2
a
n
+
1
a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}
a
1
=
2007
,
a
n
+
1
=
a
n
+
1
a
n
2
for
n
≥
1.
n \ge 1.
n
≥
1.
Prove that
[
a
n
]
=
2007
−
n
[a_{n}] =2007-n
[
a
n
]
=
2007
−
n
for
0
≤
n
≤
1004
,
0 \le n \le 1004,
0
≤
n
≤
1004
,
where
[
x
]
[x]
[
x
]
denotes the largest integer no larger than
x
.
x.
x
.
Find the largest interger
Let
n
n
n
be a positive integer and
[
n
]
=
a
.
[ \ n ] = a.
[
n
]
=
a
.
Find the largest integer
n
n
n
such that the following two conditions are satisfied:
(
1
)
(1)
(
1
)
n
n
n
is not a perfect square;
(
2
)
(2)
(
2
)
a
3
a^{3}
a
3
divides
n
2
n^{2}
n
2
.
2
2
Hide problems
Triangle Inequality.
Let
a
,
b
,
c
a,\, b,\, c
a
,
b
,
c
be side lengths of a triangle and
a
+
b
+
c
=
3
a+b+c = 3
a
+
b
+
c
=
3
. Find the minimum of
a
2
+
b
2
+
c
2
+
4
a
b
c
3
a^{2}+b^{2}+c^{2}+\frac{4abc}{3}
a
2
+
b
2
+
c
2
+
3
4
ab
c
Function equation.
Let
f
f
f
be a function given by
f
(
x
)
=
lg
(
x
+
1
)
−
1
2
⋅
log
3
x
f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x
f
(
x
)
=
l
g
(
x
+
1
)
−
2
1
⋅
lo
g
3
x
. a) Solve the equation
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
. b) Find the number of the subsets of the set
{
n
∣
f
(
n
2
−
214
n
−
1998
)
≥
0
,
n
∈
Z
}
.
\{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.
{
n
∣
f
(
n
2
−
214
n
−
1998
)
≥
0
,
n
∈
Z
}
.
1
2
Hide problems
Perpendicularity.
Let
A
B
C
ABC
A
BC
be acute triangle. The circle with diameter
A
B
AB
A
B
intersects
C
A
,
C
B
CA,\, CB
C
A
,
CB
at
M
,
N
,
M,\, N,
M
,
N
,
respectively. Draw
C
T
⊥
A
B
CT\perp AB
CT
⊥
A
B
and intersects above circle at
T
T
T
, where
C
C
C
and
T
T
T
lie on the same side of
A
B
AB
A
B
.
S
S
S
is a point on
A
N
AN
A
N
such that
B
T
=
B
S
BT = BS
BT
=
BS
. Prove that
B
S
⊥
S
C
BS\perp SC
BS
⊥
SC
.
Inequality with angles
Let
α
\alpha
α
,
β
\beta
β
be acute angles. Find the maximum value of
(
1
−
tan
α
tan
β
)
2
cot
α
+
cot
β
\frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}
cot
α
+
cot
β
(
1
−
tan
α
tan
β
)
2