MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
2004 National High School Mathematics League
2004 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
Hide problems
Vieta Theorem
α
,
β
\alpha,\beta
α
,
β
are two different solutions to the equation
4
x
2
−
4
t
x
+
1
=
0
(
t
∈
R
)
4x^2-4tx+1=0(t\in\mathbb{R})
4
x
2
−
4
t
x
+
1
=
0
(
t
∈
R
)
, the domain of definition of the function
f
(
x
)
=
2
x
−
t
x
2
+
1
f(x)=\frac{2x-t}{x^2+1}
f
(
x
)
=
x
2
+
1
2
x
−
t
is
<
a
c
l
a
s
s
=
′
l
a
t
e
x
−
h
y
p
e
r
l
i
n
k
′
h
r
e
f
=
′
α
<
β
′
>
α
,
β
<
/
a
>
<a class='latex-hyperlink' href='\alpha<\beta'>\alpha,\beta</a>
<
a
c
l
a
ss
=
′
l
a
t
e
x
−
h
y
p
er
l
in
k
′
h
re
f
=
′
α
<
β
′
>
α
,
β
<
/
a
>
. (a) Find
g
(
t
)
=
max
f
(
x
)
−
min
f
(
x
)
g(t)=\max f(x)-\min f(x)
g
(
t
)
=
max
f
(
x
)
−
min
f
(
x
)
. (b) Prove: for
u
i
∈
(
0
,
π
2
)
(
i
=
1
,
2
,
3
)
u_i\in\left(0,\frac{\pi}{2}\right)(i=1,2,3)
u
i
∈
(
0
,
2
π
)
(
i
=
1
,
2
,
3
)
, if
sin
u
1
+
sin
u
2
+
sin
u
3
=
1
\sin u_1+\sin u_2+\sin u_3=1
sin
u
1
+
sin
u
2
+
sin
u
3
=
1
, then
1
g
(
tan
u
1
)
+
1
g
(
tan
u
2
)
+
1
g
(
tan
u
3
)
<
3
4
6
\frac{1}{g(\tan u_1)}+\frac{1}{g(\tan u_2)}+\frac{1}{g(\tan u_3)}<\frac{3}{4}\sqrt6
g
(
t
a
n
u
1
)
1
+
g
(
t
a
n
u
2
)
1
+
g
(
t
a
n
u
3
)
1
<
4
3
6
.
14
1
Hide problems
Analytic Geometry
Three points
A
(
0
,
4
3
)
,
B
(
−
1
,
0
)
,
C
(
1
,
0
)
A\left(0,\frac{4}{3}\right),B(-1,0),C(1,0)
A
(
0
,
3
4
)
,
B
(
−
1
,
0
)
,
C
(
1
,
0
)
are given. The distance from
P
P
P
to line
B
C
BC
BC
is the geometric mean of that from
P
P
P
to lines
A
B
AB
A
B
and
A
C
AC
A
C
. (a) Find the path equation of point
P
P
P
. (b) If line
L
L
L
passes
D
D
D
(
D
D
D
is the incenter of
△
A
B
C
\triangle ABC
△
A
BC
), and it has three common points with the path of
P
P
P
, find the range value of slope
k
k
k
of line
L
L
L
.
13
1
Hide problems
An Interesting Game
A game about passing barriers rules that in the
n
n
n
th barrier, you need to throw a dice for
n
n
n
times. If the sum of points you get is larger than
2
n
2^n
2
n
, then you can pass this barrier. (a) How many barriers can you pass at most? (b) Find the probablity of passing the first three barriers.
12
1
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Analytic Geometry
In rectangular coordinate system, give two points
M
(
−
1
,
2
)
,
N
(
1
,
4
)
M(-1,2),N(1,4)
M
(
−
1
,
2
)
,
N
(
1
,
4
)
,
P
P
P
is a moving point on
x
x
x
-axis, when
∠
M
P
N
\angle MPN
∠
MPN
takes its maximum value, the
x
x
x
-axis of
P
P
P
is________.
11
1
Hide problems
Sequence
A sequence
a
0
,
a
1
,
a
2
,
⋯
,
a
n
,
⋯
a_0,a_1,a_2,\cdots,a_n,\cdots
a
0
,
a
1
,
a
2
,
⋯
,
a
n
,
⋯
satisfies that
a
0
=
3
a_0=3
a
0
=
3
, and
(
3
−
a
n
−
1
)
(
6
+
a
n
)
=
18
(3-a_{n-1})(6+a_n)=18
(
3
−
a
n
−
1
)
(
6
+
a
n
)
=
18
, then the value of
∑
i
=
0
n
1
a
i
\sum_{i=0}^{n}\frac{1}{a_i}
∑
i
=
0
n
a
i
1
is________.
10
1
Hide problems
Number Theory
p
p
p
is a give odd prime, if
k
2
−
p
k
\sqrt{k^2-pk}
k
2
−
p
k
is a positive integer, then the value of positive integer
k
k
k
is________.
9
1
Hide problems
3D Geometry
In cube
A
B
C
D
−
A
1
B
1
C
1
D
1
ABCD-A_1B_1C_1D_1
A
BC
D
−
A
1
B
1
C
1
D
1
, the degree of dihedral angle
A
−
B
D
1
−
A
1
A-BD_1-A_1
A
−
B
D
1
−
A
1
is________.
8
1
Hide problems
Functional Equation
Function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
, satisfies that
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
, and
f
(
x
y
+
1
)
=
f
(
x
)
f
(
y
)
−
f
(
y
)
−
x
+
2
f(xy+1)=f(x)f(y)-f(y)-x+2
f
(
x
y
+
1
)
=
f
(
x
)
f
(
y
)
−
f
(
y
)
−
x
+
2
, then
f
(
x
)
=
f(x)=
f
(
x
)
=
________.
7
1
Hide problems
Find the area
In rectangular coordinate system, the area which is surrounded by the figure of
f
(
x
)
=
a
sin
a
x
+
cos
a
x
(
a
>
0
)
f(x)=a\sin ax+\cos ax(a>0)
f
(
x
)
=
a
sin
a
x
+
cos
a
x
(
a
>
0
)
on a complete period and the figure of
g
(
x
)
=
a
2
+
1
g(x)=\sqrt{a^2+1}
g
(
x
)
=
a
2
+
1
is________.
6
1
Hide problems
3D Geometry
Shaft section of a circular cone with vertex
P
P
P
is an isosceles right triangle.
A
A
A
is a point on the circle of the bottom surface, while
B
B
B
is a point inside the circle,
O
O
O
is the center of the circle. If
A
B
⊥
O
B
AB\perp OB
A
B
⊥
OB
at
B
B
B
,
O
H
⊥
P
B
OH\perp PB
O
H
⊥
PB
at
H
H
H
,
P
A
=
4
PA=4
P
A
=
4
,
C
C
C
is the midpoint of
P
A
PA
P
A
, then when the volume of triangular pyramid
O
−
H
P
C
O-HPC
O
−
H
PC
takes its maximum value, the length of
O
B
OB
OB
is
(A)
5
3
(B)
2
5
3
(C)
6
3
(D)
2
6
3
\text{(A)}\frac{\sqrt5}{3}\qquad\text{(B)}\frac{2\sqrt5}{3}\qquad\text{(C)}\frac{\sqrt6}{3}\qquad\text{(D)}\frac{2\sqrt6}{3}\qquad
(A)
3
5
(B)
3
2
5
(C)
3
6
(D)
3
2
6
5
1
Hide problems
Count the Number
For a 3-digit-number
n
=
a
b
c
‾
n=\overline{abc}
n
=
ab
c
, if
a
,
b
,
c
a,b,c
a
,
b
,
c
can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is
(A)
45
(B)
81
(C)
165
(D)
216
\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216
(A)
45
(B)
81
(C)
165
(D)
216
4
1
Hide problems
Vector Problem
O
O
O
is a point inside
△
A
B
C
\triangle ABC
△
A
BC
, and
O
A
→
+
2
O
B
→
+
3
O
C
→
=
0
→
\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}
O
A
+
2
OB
+
3
OC
=
0
, then the ratio of the area of
△
A
B
C
\triangle ABC
△
A
BC
to
△
A
O
C
\triangle AOC
△
A
OC
is
(A)
2
(B)
3
2
(C)
3
(D)
5
3
\text{(A)}2\qquad\text{(B)}\frac{3}{2}\qquad\text{(C)}3\qquad\text{(D)}\frac{5}{3}
(A)
2
(B)
2
3
(C)
3
(D)
3
5
3
2
Hide problems
Solve the Ineqlity
The solution set to the inequality
log
2
x
−
1
+
1
2
log
1
2
x
3
+
2
>
0
\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0
lo
g
2
x
−
1
+
2
1
lo
g
2
1
x
3
+
2
>
0
is
(A)
[
2
,
3
)
(B)
(
2
,
3
]
(C)
[
2
,
4
)
(D)
(
2
,
4
]
\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]
(A)
[
2
,
3
)
(B)
(
2
,
3
]
(C)
[
2
,
4
)
(D)
(
2
,
4
]
Number Theory
For integer
n
≥
4
n\geq4
n
≥
4
, find the smallest 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, \cdots, m+n-1\}
{
m
,
m
+
1
,
⋯
,
m
+
n
−
1
}
there are at least three elements that are relatively prime .
2
2
Hide problems
Two Sets
Two sets
M
=
{
(
x
,
y
)
∣
x
2
+
2
y
2
=
3
}
,
N
=
{
(
x
,
y
)
∣
y
=
m
x
+
b
}
M=\{(x,y)|x^2+2y^2=3\},N=\{(x,y)|y=mx+b\}
M
=
{(
x
,
y
)
∣
x
2
+
2
y
2
=
3
}
,
N
=
{(
x
,
y
)
∣
y
=
m
x
+
b
}
. For all
m
∈
R
m\in\mathbb{R}
m
∈
R
,
M
∩
N
≠
∅
M\cap N\neq\varnothing
M
∩
N
=
∅
, then the range value of
b
b
b
is
(A)
[
−
6
2
,
6
2
]
(B)
(
−
6
2
,
6
2
)
(C)
(
−
2
3
3
,
2
3
3
]
(D)
[
−
2
3
3
,
2
3
3
]
\text{(A)}\left[-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right]\qquad\text{(B)}\left(-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right)\qquad\text{(C)}\left(-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]\qquad\text{(D)}\left[-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]
(A)
[
−
2
6
,
2
6
]
(B)
(
−
2
6
,
2
6
)
(C)
(
−
3
2
3
,
3
2
3
]
(D)
[
−
3
2
3
,
3
2
3
]
Too Many Numbers
In rectangular coordinate system, define two sequences of points:
(
A
n
)
(A_n)
(
A
n
)
on the positive half of the
y
y
y
-axis and
(
B
n
)
(B_n)
(
B
n
)
on the curve
y
=
2
x
(
x
≥
0
)
y=\sqrt{2x}(x\geq0)
y
=
2
x
(
x
≥
0
)
satisfy that
∣
O
A
n
∣
=
∣
O
B
n
∣
=
1
n
|OA_n|=|OB_n|=\frac{1}{n}
∣
O
A
n
∣
=
∣
O
B
n
∣
=
n
1
.
a
n
a_n
a
n
is the
x
x
x
-intercept of line
A
n
B
n
A_nB_n
A
n
B
n
, and the
x
x
x
-axis of
B
n
B_n
B
n
is
b
n
b_n
b
n
,
n
∈
Z
+
n\in\mathbb{Z}_+
n
∈
Z
+
. Prove: (a)
a
n
>
a
n
+
1
>
4
,
n
∈
Z
+
a_n>a_{n+1}>4,n\in\mathbb{Z}_+
a
n
>
a
n
+
1
>
4
,
n
∈
Z
+
; (b) There exists
n
0
∈
Z
+
n_0\in\mathbb{Z}_+
n
0
∈
Z
+
, such that
∀
n
>
n
0
\forall 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}+\cdots +\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
2
Hide problems
An Equation
If the equation
x
2
+
4
x
cos
θ
+
cot
θ
=
0
x^2+4x\cos\theta+\cot\theta=0
x
2
+
4
x
cos
θ
+
cot
θ
=
0
has a repeated root, where
θ
\theta
θ
is an acute angle, then the radian of
θ
\theta
θ
is
(A)
π
6
(B)
π
12
or
5
π
12
(C)
π
6
or
5
π
12
(D)
π
12
\text{(A)}\frac{\pi}{6}\qquad\text{(B)}\frac{\pi}{12}\text{ or }\frac{5\pi}{12}\qquad\text{(C)}\frac{\pi}{6}\text{ or }\frac{5\pi}{12}\qquad\text{(D)}\frac{\pi}{12}
(A)
6
π
(B)
12
π
or
12
5
π
(C)
6
π
or
12
5
π
(D)
12
π
Geometry
In acute triangle
A
B
C
ABC
A
BC
, point
H
H
H
is the intersection point of heights
C
E
CE
CE
on side
A
B
AB
A
B
and
B
D
BD
B
D
on side
A
C
AC
A
C
. A circle with diameter
D
E
DE
D
E
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
K
K
K
. If
B
C
=
25
,
B
D
=
20
,
B
E
=
7
BC=25,BD=20, BE=7
BC
=
25
,
B
D
=
20
,
BE
=
7
, find the length of
A
K
AK
A
K
.