MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1994 National High School Mathematics League
1994 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(12)
12
1
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The Minumum Positive Value
95 numbers
a
1
,
a
2
,
⋯
,
a
95
a_1,a_2,\cdots,a_{95}
a
1
,
a
2
,
⋯
,
a
95
are either
1
1
1
or
−
1
-1
−
1
. Then the minumum positive value of
∑
1
≤
i
<
j
≤
95
a
i
a
j
\sum_{1\leq i<j\leq95}a_i a_j
∑
1
≤
i
<
j
≤
95
a
i
a
j
is________.
11
1
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12 Intersection Angles
Intersections between a plane and 12 edges of a cube are all
α
\alpha
α
, then
sin
α
=
\sin\alpha=
sin
α
=
________.
10
1
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The Maximum Value
If
0
<
θ
<
π
0<\theta<\pi
0
<
θ
<
π
, then the maximum value of
sin
θ
2
(
1
+
cos
θ
)
\sin\frac{\theta}{2}(1+\cos\theta)
sin
2
θ
(
1
+
cos
θ
)
is________.
9
1
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The Number of Points
Point Sets
A
=
{
(
x
,
y
)
∣
(
x
−
3
)
2
+
(
y
−
4
)
2
≤
(
5
2
)
2
}
,
B
=
{
(
x
,
y
)
∣
(
x
−
4
)
2
+
(
y
−
5
)
2
>
(
5
2
)
2
}
A=\{(x,y)|(x-3)^2+(y-4)^2\leq\left( \frac{5}{2}\right)^2\},B=\{(x,y)|(x-4)^2+(y-5)^2>\left( \frac{5}{2}\right)^2\}
A
=
{(
x
,
y
)
∣
(
x
−
3
)
2
+
(
y
−
4
)
2
≤
(
2
5
)
2
}
,
B
=
{(
x
,
y
)
∣
(
x
−
4
)
2
+
(
y
−
5
)
2
>
(
2
5
)
2
}
, then the number of integral points in
A
∩
B
A\cap B
A
∩
B
is________.
8
1
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Trigonometry
x
,
y
∈
[
−
π
4
,
π
4
]
,
a
∈
R
x,y\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right],a\in\mathbb{R}
x
,
y
∈
[
−
4
π
,
4
π
]
,
a
∈
R
. If
x
3
+
sin
x
−
2
a
=
0
,
4
y
3
+
sin
y
cos
y
+
a
=
0
x^3+\sin x-2a=0,4y^3+\sin y \cos y+a=0
x
3
+
sin
x
−
2
a
=
0
,
4
y
3
+
sin
y
cos
y
+
a
=
0
, then
cos
(
x
+
2
y
)
=
\cos (x+2y)=
cos
(
x
+
2
y
)
=
________.
7
1
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Directed Line Segment
A directed line segment, starting point is
P
(
−
1
,
1
)
P(-1,1)
P
(
−
1
,
1
)
, finishing point is
Q
(
2
,
2
)
Q(2,2)
Q
(
2
,
2
)
. If line
l
:
x
+
m
y
+
m
=
0
l:x+my+m=0
l
:
x
+
m
y
+
m
=
0
intersects
P
Q
PQ
PQ
at its extended line, then the range value of
m
m
m
is________.
6
1
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Rectangular Coordinate System Problem
In rectangular coordinate system, the equation
∣
x
+
y
∣
2
a
+
∣
x
−
y
∣
2
b
=
1
\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1
2
a
∣
x
+
y
∣
+
2
b
∣
x
−
y
∣
=
1
(
a
,
b
a,b
a
,
b
are different positive numbers) refers to
(A)
\text{(A)}
(A)
a triangle
(B)
\text{(B)}
(B)
a square
(C)
\text{(C)}
(C)
rectangle, not square
(D)
\text{(D)}
(D)
rhombus, not square
5
1
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Pyramid Problem
In regular
n
n
n
-regular pyramid, the range value of dihedral angle of two adjacent sides is
(A)
(
n
−
2
n
π
,
π
)
(B)
(
n
−
1
n
π
,
π
)
(C)
(
0
,
π
2
)
(D)
(
n
−
2
n
π
,
n
−
1
n
π
)
\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)
(A)
(
n
n
−
2
π
,
π
)
(B)
(
n
n
−
1
π
,
π
)
(C)
(
0
,
2
π
)
(D)
(
n
n
−
2
π
,
n
n
−
1
π
)
4
2
Hide problems
Order the Numbers
0
<
b
<
1
,
0
<
a
<
π
4
0<b<1,0<a<\frac{\pi}{4}
0
<
b
<
1
,
0
<
a
<
4
π
,
x
=
(
sin
a
)
log
b
sin
a
,
y
=
(
cos
a
)
log
b
cos
a
,
z
=
(
sin
a
)
log
b
cos
a
x=(\sin a)^{\log_{b}\sin a},y=(\cos a)^{\log_{b}\cos a},z=(\sin a)^{\log_{b}\cos a}
x
=
(
sin
a
)
l
o
g
b
s
i
n
a
,
y
=
(
cos
a
)
l
o
g
b
c
o
s
a
,
z
=
(
sin
a
)
l
o
g
b
c
o
s
a
. Then the order of
x
,
y
,
z
x,y,z
x
,
y
,
z
is
(A)
x
<
z
<
y
(B)
y
<
z
<
x
(C)
z
<
x
<
y
(D)
x
<
y
<
z
\text{(A)}x<z<y\qquad\text{(B)}y<z<x\qquad\text{(C)}z<x<y\qquad\text{(D)}x<y<z
(A)
x
<
z
<
y
(B)
y
<
z
<
x
(C)
z
<
x
<
y
(D)
x
<
y
<
z
1994 Points
Give a point set on a plane
P
=
{
P
1
,
P
2
,
⋯
,
P
1994
}
P=\{P_1,P_2,\cdots,P_{1994}\}
P
=
{
P
1
,
P
2
,
⋯
,
P
1994
}
. Any three points in
P
P
P
are not colinear. Divide points in
P
P
P
into
83
83
83
groups, in each group there are at least three points, and any point exactly belongs to one group. Connect any two points in the same group with a line segment, but we do not connect points not in the same group. Now we get a figure
G
G
G
. Note the number of triangles in
G
G
G
:
m
(
G
)
m(G)
m
(
G
)
. (a) Find the minumum value of
m
(
G
)
m(G)
m
(
G
)
:
m
0
m_0
m
0
. (b)
G
∗
G*
G
∗
is one of the figures that
m
(
G
)
=
m
0
m(G)=m_0
m
(
G
)
=
m
0
, color the line segments in
G
∗
G*
G
∗
with four colors. Prove that there exists a proper way, satisfying that no triangle is made up of three sides in the same color.
3
2
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Sum of Sequence
Sequence
(
a
n
)
(a_n)
(
a
n
)
satisfies that
3
a
n
+
1
+
a
n
=
4
(
n
≥
1
)
,
a
1
=
9
3a_{n+1}+a_n=4(n\geq1),a_1=9
3
a
n
+
1
+
a
n
=
4
(
n
≥
1
)
,
a
1
=
9
, let
S
n
=
∑
i
=
1
n
a
i
S_n=\sum_{i=1}^{n}a_i
S
n
=
∑
i
=
1
n
a
i
, then the minumum value of
n
n
n
such that
∣
S
n
−
n
−
6
∣
<
1
125
|S_n-n-6|<\frac{1}{125}
∣
S
n
−
n
−
6∣
<
125
1
is
(A)
5
(B)
6
(C)
7
(D)
8
\text{(A)}5\qquad\text{(B)}6\qquad\text{(C)}7\qquad\text{(D)}8
(A)
5
(B)
6
(C)
7
(D)
8
Geometry
Circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
is
⊙
O
\odot O
⊙
O
, incentre of
△
A
B
C
\triangle ABC
△
A
BC
is
I
I
I
.
∠
B
=
6
0
∘
.
∠
A
<
∠
C
\angle B=60^{\circ}.\angle A<\angle C
∠
B
=
6
0
∘
.∠
A
<
∠
C
. Bisector of outer angle
∠
A
\angle A
∠
A
intersects
⊙
O
\odot O
⊙
O
at
E
E
E
. Prove: (a)
I
O
=
A
E
IO=AE
I
O
=
A
E
. (b) The radius of
⊙
O
\odot O
⊙
O
is
R
R
R
, then
2
R
<
I
O
+
I
A
+
I
C
<
(
1
+
3
)
R
2R<IO+IA+IC<(1+\sqrt3)R
2
R
<
I
O
+
I
A
+
I
C
<
(
1
+
3
)
R
.
2
2
Hide problems
Two Statements
Give two statements: (1)
a
,
b
,
c
a,b,c
a
,
b
,
c
are complex numbers, if
a
2
+
b
2
>
c
2
a^2+b^2>c^2
a
2
+
b
2
>
c
2
, then
a
2
+
b
2
−
c
2
>
0
a^2+b^2-c^2>0
a
2
+
b
2
−
c
2
>
0
. (2)
a
,
b
,
c
a,b,c
a
,
b
,
c
are complex numbers, if
a
2
+
b
2
−
c
2
>
0
a^2+b^2-c^2>0
a
2
+
b
2
−
c
2
>
0
, then
a
2
+
b
2
>
c
2
a^2+b^2>c^2
a
2
+
b
2
>
c
2
. Then, which is true?
(A)
\text{(A)}
(A)
(1) is correct, (2) is correct as well
(B)
\text{(B)}
(B)
(1) is correct, (2) is incorrect
(C)
\text{(C)}
(C)
(1) is incorrect, (2) is incorrect as well
(D)
\text{(D)}
(D)
(1) is incorrect, (2) is correct
The 1000th Number
Find the 1000th number (from small to large) that is coprime to
105
105
105
.
1
2
Hide problems
Inequality
a
,
b
,
c
a,b,c
a
,
b
,
c
are real numbers. The sufficient and necessary condition of
∀
x
∈
R
,
a
sin
x
+
b
cos
x
+
c
>
0
\forall x\in\mathbb{R},a\sin x+b\cos x+c>0
∀
x
∈
R
,
a
sin
x
+
b
cos
x
+
c
>
0
is
(A)
\text{(A)}
(A)
a
=
b
=
0
,
c
>
0
a=b=0,c>0
a
=
b
=
0
,
c
>
0
(B)
\text{(B)}
(B)
a
2
+
b
2
=
c
\sqrt{a^2+b^2}=c
a
2
+
b
2
=
c
(C)
\text{(C)}
(C)
a
2
+
b
2
<
c
\sqrt{a^2+b^2}<c
a
2
+
b
2
<
c
(D)
\text{(D)}
(D)
a
2
+
b
2
>
c
\sqrt{a^2+b^2}>c
a
2
+
b
2
>
c
Complex Number
In the equation
x
2
+
z
1
x
+
z
2
+
m
=
0
x^2+z_1x+z_2+m=0
x
2
+
z
1
x
+
z
2
+
m
=
0
,
z
1
,
z
2
,
m
z_1,z_2,m
z
1
,
z
2
,
m
are complex numbers, and
z
1
2
−
4
z
2
=
16
+
20
i
z_1^2-4z_2=16+20\text{i}
z
1
2
−
4
z
2
=
16
+
20
i
. Two roots of the equations are
α
,
β
\alpha,\beta
α
,
β
. If
∣
α
−
β
∣
=
2
7
|\alpha-\beta|=2\sqrt7
∣
α
−
β
∣
=
2
7
, find the maximum and minumum value of
∣
m
∣
|m|
∣
m
∣
.