MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2004 China Second Round Olympiad
2004 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
Relatively prime elements
For integer
n
≥
4
n\ge 4
n
≥
4
, find the minimal integer
f
(
n
)
f(n)
f
(
n
)
, such that for any positive integer
m
m
m
, in any subset with
f
(
n
)
f(n)
f
(
n
)
elements of the set
m
,
m
+
1
,
…
,
m
+
n
+
1
{m, m+1, \ldots, m+n+1}
m
,
m
+
1
,
…
,
m
+
n
+
1
there are at least
3
3
3
relatively prime elements.
2
1
Hide problems
Sequences of points
In a planar rectangular coordinate system, a sequence of points
A
n
{A_n}
A
n
on the positive half of the y-axis and a sequence of points
B
n
{B_n}
B
n
on the curve
y
=
2
x
y=\sqrt{2x}
y
=
2
x
(
x
≥
0
)
(x\ge0)
(
x
≥
0
)
satisfy the condition
∣
O
A
n
∣
=
∣
O
B
n
∣
=
1
n
|OA_n|=|OB_n|=\frac{1}{n}
∣
O
A
n
∣
=
∣
O
B
n
∣
=
n
1
. The x-intercept of line
A
n
B
n
A_nB_n
A
n
B
n
is
a
n
a_n
a
n
, and the x-coordinate of point
B
n
B_n
B
n
is
b
n
b_n
b
n
,
n
∈
N
n\in\mathbb{N}
n
∈
N
. Prove that (1)
a
n
>
a
n
+
1
>
4
a_n>a_{n+1}>4
a
n
>
a
n
+
1
>
4
,
n
∈
N
n\in\mathbb{N}
n
∈
N
; (2) There is
n
0
∈
N
n_0\in\mathbb{N}
n
0
∈
N
, such that for any
n
>
n
0
n>n_0
n
>
n
0
,
b
2
b
1
+
b
3
b
2
+
…
+
b
n
b
n
−
1
+
b
n
+
1
b
n
<
n
−
2004
\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004
b
1
b
2
+
b
2
b
3
+
…
+
b
n
−
1
b
n
+
b
n
b
n
+
1
<
n
−
2004
.
1
1
Hide problems
acute triangle
In an acute triangle
A
B
C
ABC
A
BC
, point
H
H
H
is the intersection point of altitude
C
E
CE
CE
to
A
B
AB
A
B
and altitude
B
D
BD
B
D
to
A
C
AC
A
C
. A circle with
D
E
DE
D
E
as its diameter intersects
A
B
AB
A
B
and
A
C
AC
A
C
at
F
F
F
and
G
G
G
, respectively.
F
G
FG
FG
and
A
H
AH
A
H
intersect at point
K
K
K
. If
B
C
=
25
BC=25
BC
=
25
,
B
D
=
20
BD=20
B
D
=
20
, and
B
E
=
7
BE=7
BE
=
7
, find the length of
A
K
AK
A
K
.