MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1995 National High School Mathematics League
1995 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(12)
12
1
Hide problems
How Many Numbers?
Set
M
=
{
1
,
2
,
⋯
,
1995
}
M=\{1,2,\cdots,1995\}
M
=
{
1
,
2
,
⋯
,
1995
}
.
A
A
A
is a subset of
M
M
M
such that
∀
x
∈
A
,
15
x
∉
A
\forall x\in A,15x\not\in A
∀
x
∈
A
,
15
x
∈
A
. Then the maximum
∣
A
∣
|A|
∣
A
∣
is________.
11
1
Hide problems
Quadrangular Pyramid Counting Problem
Color the vertexes of a quadrangular pyramid in one color, satisfying that two end points of any edge are in different colors. We have only 5 colors, then the number of ways coloring the quadrangular pyramid is________.
10
1
Hide problems
The Number of Integral Points
The number of integral points satisfy
{
y
≤
3
x
y
≥
x
3
x
+
y
≥
100
\begin{cases} y\leq 3x\\ y\geq \frac{x}{3}\\ x+y\geq100 \end{cases}
⎩
⎨
⎧
y
≤
3
x
y
≥
3
x
x
+
y
≥
100
on the coordinate plane is________.
9
1
Hide problems
Gaussian Equation
The number of real roots of the equation
lg
2
x
−
[
lg
x
]
−
2
=
0
\lg^2x-[\lg x]-2=0
l
g
2
x
−
[
l
g
x
]
−
2
=
0
is________.
8
1
Hide problems
Two Volumes
Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.
7
1
Hide problems
Conjugate Complex Numbers
α
,
β
\alpha,\beta
α
,
β
are conjugate complex numbers. If
∣
α
−
β
∣
=
2
3
|\alpha-\beta|=2\sqrt3
∣
α
−
β
∣
=
2
3
,
α
β
2
\frac{\alpha}{\beta^2}
β
2
α
is a real number, then
∣
α
∣
=
|\alpha|=
∣
α
∣
=
________.
6
1
Hide problems
Regular Triangular Pyramid
O
O
O
is the center of the bottom surface of regular triangular pyramid
P
−
A
B
C
P-ABC
P
−
A
BC
. A plane passes
O
O
O
intersects line segment
P
C
PC
PC
at
S
S
S
, intersects the extended line of
P
A
,
P
B
PA,PB
P
A
,
PB
at
Q
,
R
Q,R
Q
,
R
, then
1
∣
P
Q
∣
+
1
∣
P
R
∣
+
1
∣
P
S
∣
\frac{1}{|PQ|}+\frac{1}{|PR|}+\frac{1}{|PS|}
∣
PQ
∣
1
+
∣
PR
∣
1
+
∣
PS
∣
1
(A)
\text{(A)}
(A)
has a maximum value, but no minumum value
(B)
\text{(B)}
(B)
has a minumum value, but no maximum value
(C)
\text{(C)}
(C)
has both minumum value and maximum value (different)
(D)
\text{(D)}
(D)
is a fixed value
5
1
Hide problems
Four Numbers
The order of
log
sin
1
cos
1
,
log
sin
1
tan
1
,
log
cos
1
sin
1
,
log
cos
1
tan
1
\log_{\sin1}\cos1,\log_{\sin1}\tan1,\log_{\cos1}\sin1,\log_{\cos1}\tan1
lo
g
s
i
n
1
cos
1
,
lo
g
s
i
n
1
tan
1
,
lo
g
c
o
s
1
sin
1
,
lo
g
c
o
s
1
tan
1
is (form small to large)
(A)
log
sin
1
cos
1
<
log
cos
1
sin
1
<
log
sin
1
tan
1
<
log
cos
1
tan
1
\text{(A)}\log_{\sin1}\cos1<\log_{\cos1}\sin1<\log_{\sin1}\tan1<\log_{\cos1}\tan1
(A)
lo
g
s
i
n
1
cos
1
<
lo
g
c
o
s
1
sin
1
<
lo
g
s
i
n
1
tan
1
<
lo
g
c
o
s
1
tan
1
(B)
log
cos
1
sin
1
<
log
cos
1
tan
1
<
log
sin
1
cos
1
<
log
sin
1
tan
1
\text{(B)}\log_{\cos1}\sin1<\log_{\cos1}\tan1<\log_{\sin1}\cos1<\log_{\sin1}\tan1
(B)
lo
g
c
o
s
1
sin
1
<
lo
g
c
o
s
1
tan
1
<
lo
g
s
i
n
1
cos
1
<
lo
g
s
i
n
1
tan
1
(C)
log
sin
1
tan
1
<
log
cos
1
tan
1
<
log
cos
1
sin
1
<
log
sin
1
cos
1
\text{(C)}\log_{\sin1}\tan1<\log_{\cos1}\tan1<\log_{\cos1}\sin1<\log_{\sin1}\cos1
(C)
lo
g
s
i
n
1
tan
1
<
lo
g
c
o
s
1
tan
1
<
lo
g
c
o
s
1
sin
1
<
lo
g
s
i
n
1
cos
1
(D)
log
cos
1
tan
1
<
log
sin
1
tan
1
<
log
sin
1
cos
1
<
log
cos
1
sin
1
\text{(D)}\log_{\cos1}\tan1<\log_{\sin1}\tan1<\log_{\sin1}\cos1<\log_{\cos1}\sin1
(D)
lo
g
c
o
s
1
tan
1
<
lo
g
s
i
n
1
tan
1
<
lo
g
s
i
n
1
cos
1
<
lo
g
c
o
s
1
sin
1
4
2
Hide problems
Roots of the Equation
Equation
∣
x
−
2
n
∣
=
k
x
(
n
∈
Z
+
)
|x-2n|=k\sqrt{x}(n\in\mathbb{Z}_+)
∣
x
−
2
n
∣
=
k
x
(
n
∈
Z
+
)
has two different real roots on
(
2
n
−
1
,
2
n
+
1
]
(2n-1,2n+1]
(
2
n
−
1
,
2
n
+
1
]
, then the range value of
k
k
k
is
(A)
k
>
0
(B)
0
<
k
≤
1
2
n
+
1
(C)
1
2
n
+
1
<
k
≤
1
2
n
+
1
\text{(A)}k>0\qquad\text{(B)}0<k\leq\frac{1}{\sqrt{2n+1}}\qquad\text{(C)}\frac{1}{2n+1}<k\leq\frac{1}{\sqrt{2n+1}}
(A)
k
>
0
(B)
0
<
k
≤
2
n
+
1
1
(C)
2
n
+
1
1
<
k
≤
2
n
+
1
1
(D)
\text{(D)}
(D)
none above
Color all points
Color all points on a plane in red or blue. Prove that there exists two similar triangles, their similarity ratio is
1995
1995
1995
, and apexes of both triangles are in the same color.
3
2
Hide problems
Are You an Excellent Boy?
If a person A is taller or heavier than another peoson B, then we note that A is not worse than B. In 100 persons, if someone is not worse than other 99 people, we call him excellent boy. What's the maximum value of the number of excellent boys?
(A)
1
(B)
2
(C)
50
(D)
100
\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}50\qquad\text{(D)}100
(A)
1
(B)
2
(C)
50
(D)
100
Inscribed Circle
Inscribed Circle of rhombus
A
B
C
D
ABCD
A
BC
D
touches
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
at
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
.
l
1
,
l
2
l_1,l_2
l
1
,
l
2
are two lines that are tangent to the circle.
l
1
∩
A
B
=
M
,
l
1
∩
B
C
=
N
,
l
2
∩
C
D
=
P
,
l
2
∩
D
A
=
Q
l_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q
l
1
∩
A
B
=
M
,
l
1
∩
BC
=
N
,
l
2
∩
C
D
=
P
,
l
2
∩
D
A
=
Q
. Prove that
M
Q
/
/
N
P
MQ/\! /NP
MQ
/
/
NP
.
2
2
Hide problems
Complex Plane
Complex numbers of apexes of 20-regular polygon inscribed to unit circle refer to are
Z
1
,
Z
2
,
⋯
,
Z
20
Z_1,Z_2,\cdots,Z_{20}
Z
1
,
Z
2
,
⋯
,
Z
20
on complex plane. Then the number of points in
Z
1
1995
,
Z
2
1995
,
⋯
,
Z
20
1995
Z_1^{1995},Z_2^{1995},\cdots,Z_{20}^{1995}
Z
1
1995
,
Z
2
1995
,
⋯
,
Z
20
1995
refer to is
(A)
4
(B)
5
(C)
10
(D)
20
\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}10\qquad\text{(D)}20
(A)
4
(B)
5
(C)
10
(D)
20
Three Roots
Find all real number
p
p
p
, such that the three roots of the equation
5
x
3
−
5
(
p
+
1
)
x
2
+
(
71
p
−
1
)
x
+
1
=
66
p
5x^3-5(p+1)x^2+(71p-1)x+1=66p
5
x
3
−
5
(
p
+
1
)
x
2
+
(
71
p
−
1
)
x
+
1
=
66
p
are all positive integers.
1
2
Hide problems
Arithmetic Sequence
In arithmetic sequence
(
a
n
)
(a_n)
(
a
n
)
,
3
a
8
=
5
a
13
,
a
1
>
0
3a_8=5a_{13},a_1>0
3
a
8
=
5
a
13
,
a
1
>
0
. Define
S
n
=
∑
i
=
1
n
a
i
S_n=\sum_{i=1}^n a_i
S
n
=
∑
i
=
1
n
a
i
, then the largest number in
(
S
n
)
(S_n)
(
S
n
)
is
(A)
S
10
(B)
S
11
(C)
S
20
(D)
S
21
\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}
(A)
S
10
(B)
S
11
(C)
S
20
(D)
S
21
Family of Curves
Give a family of curves
2
(
2
sin
θ
−
cos
θ
+
3
)
x
2
−
(
8
sin
θ
+
cos
θ
+
1
)
=
0
2(2\sin\theta-\cos\theta+3)x^2-(8\sin\theta+\cos\theta+1)=0
2
(
2
sin
θ
−
cos
θ
+
3
)
x
2
−
(
8
sin
θ
+
cos
θ
+
1
)
=
0
, where
θ
\theta
θ
is a parameter. Find the maximum value of the length of the chord that
y
=
2
x
y=2x
y
=
2
x
intersects the curve.