MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1998 National High School Mathematics League
1998 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Analytic Geometry
Parabola
y
2
=
2
p
x
y^2=2px
y
2
=
2
p
x
, two fixed points
A
(
a
,
b
)
,
B
(
−
a
,
0
)
(
a
b
≠
0
,
b
2
≠
2
p
a
)
A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)
A
(
a
,
b
)
,
B
(
−
a
,
0
)
(
ab
=
0
,
b
2
=
2
p
a
)
.
M
M
M
is a point on the parabola,
A
M
AM
A
M
intersects the parabola at
M
1
M_1
M
1
,
B
M
BM
BM
intersects the parabola at
M
2
M_2
M
2
. Prove: When
M
M
M
changes, line
M
1
M
2
M_1M_2
M
1
M
2
passes a fixed point, and find the fixed point.
14
1
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Function Problem
Function
f
(
x
)
=
a
x
2
+
8
x
+
3
(
a
<
0
)
f(x)=ax^2+8x+3(a<0)
f
(
x
)
=
a
x
2
+
8
x
+
3
(
a
<
0
)
. For any given nerative number
a
a
a
, define the largest positive number
l
(
a
)
l(a)
l
(
a
)
:
∣
f
(
x
)
∣
≤
5
|f(x)|\leq5
∣
f
(
x
)
∣
≤
5
for all
x
∈
[
0
,
l
(
a
)
]
x\in[0,l(a)]
x
∈
[
0
,
l
(
a
)]
. Find the largest
l
(
a
)
l(a)
l
(
a
)
, and
a
a
a
when
l
(
a
)
l(a)
l
(
a
)
takes its maximum value.
13
1
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Complex Number
Complex number
z
=
1
−
sin
θ
+
i
cos
θ
(
π
2
<
θ
<
π
)
z=1-\sin\theta+\text{i}\cos\theta\left(\frac{\pi}{2}<\theta<\pi\right)
z
=
1
−
sin
θ
+
i
cos
θ
(
2
π
<
θ
<
π
)
, find the range value of
arg
z
‾
\arg{\overline{z}}
ar
g
z
.
12
1
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3D Geometry
In
△
A
B
C
\triangle ABC
△
A
BC
,
∠
C
=
9
0
∘
,
∠
B
=
3
0
∘
,
A
C
=
2
\angle C=90^{\circ},\angle B=30^{\circ}, AC=2
∠
C
=
9
0
∘
,
∠
B
=
3
0
∘
,
A
C
=
2
.
M
M
M
is the midpoint of
A
B
AB
A
B
. Fold up
△
A
C
M
\triangle ACM
△
A
CM
along
C
M
CM
CM
, satisfying that
∣
A
B
∣
=
2
2
|AB|=2\sqrt2
∣
A
B
∣
=
2
2
. The volume of triangular pyramid
A
−
B
C
M
A-BCM
A
−
BCM
is________.
11
1
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Analytic Geometry
If ellipse
x
2
+
4
(
y
−
a
)
2
=
4
x^2+4(y-a)^2=4
x
2
+
4
(
y
−
a
)
2
=
4
and parabola
x
2
=
2
y
x^2=2y
x
2
=
2
y
have intersections, then the range value of
a
a
a
is________.
10
1
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Arithmetic Sequence
Arithmetic sequence with all items real, and the common difference is
4
4
4
. If the sum of the square of the first item and all items else is not more than
100
100
100
, then there are________items at most.
9
1
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Count the Number
Pick out three numbers from
0
,
1
,
⋯
,
9
0,1,\cdots,9
0
,
1
,
⋯
,
9
, their sum is an even number and not less than
10
10
10
. We have________different ways to pick numbers.
8
1
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Complex Number
Complex number
z
=
cos
θ
+
i
sin
θ
(
0
≤
θ
≤
π
)
z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)
z
=
cos
θ
+
i
sin
θ
(
0
≤
θ
≤
π
)
. Points that three complex numbers
z
,
(
1
+
i
)
z
,
2
z
‾
z,(1+\text{i})z,2\overline{z}
z
,
(
1
+
i
)
z
,
2
z
refer to on complex plane are
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
. When
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are not collinear,
P
Q
S
R
PQSR
PQSR
is a parallelogram. The longest distance between
S
S
S
and the original point is________.
7
1
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Even Function
f
(
x
)
f(x)
f
(
x
)
is an even function with period of
2
2
2
. If
f
(
x
)
=
x
1
1000
f(x)=x^{\frac{1}{1000}}
f
(
x
)
=
x
1000
1
when
x
∈
[
0
,
1
]
x\in[0,1]
x
∈
[
0
,
1
]
, then the order of
f
(
98
19
)
,
f
(
101
17
)
,
f
(
104
15
)
f\left(\frac{98}{19}\right),f\left(\frac{101}{17}\right),f\left(\frac{104}{15}\right)
f
(
19
98
)
,
f
(
17
101
)
,
f
(
15
104
)
is________(from small to large).
6
1
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Groups of Three Collinear Points
In the 27 points of a cube: 8 vertexes, 12 midpoints of edges, 6 centers of surfaces, and the center of the cube, the number of groups of three collinear points is
(A)
57
(B)
49
(C)
43
(D)
37
\text{(A)}57\qquad\text{(B)}49\qquad\text{(C)}43\qquad\text{(D)}37
(A)
57
(B)
49
(C)
43
(D)
37
5
1
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Regular Tetrahedron
In regular tetrahedron
A
B
C
D
ABCD
A
BC
D
,
E
,
F
,
G
E,F,G
E
,
F
,
G
are midpoints of
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
. Dihedral angle
C
−
F
G
−
E
C-FG-E
C
−
FG
−
E
is equal to
(A)
arcsin
6
3
(B)
π
2
+
arccos
3
3
(C)
π
2
−
arctan
2
(D)
π
−
arccot
2
2
\text{(A)}\arcsin\frac{\sqrt6}{3}\qquad\text{(B)}\frac{\pi}{2}+\arccos\frac{\sqrt3}{3}\qquad\text{(C)}\frac{\pi}{2}-\arctan{\sqrt2}\qquad\text{(D)}\pi-\text{arccot}\frac{\sqrt2}{2}
(A)
arcsin
3
6
(B)
2
π
+
arccos
3
3
(C)
2
π
−
arctan
2
(D)
π
−
arccot
2
2
4
1
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Quadratic Inequality
Statement
P
P
P
: solution set to inequalities
a
1
x
2
+
b
1
x
+
c
1
>
0
a_1x^2+b_1x+c_1>0
a
1
x
2
+
b
1
x
+
c
1
>
0
and
a
2
x
2
+
b
2
x
+
c
2
>
0
a_2x^2+b_2x+c_2>0
a
2
x
2
+
b
2
x
+
c
2
>
0
are the same; statement
Q
Q
Q
:
a
1
a
2
=
b
1
b
2
=
c
1
c
2
\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}
a
2
a
1
=
b
2
b
1
=
c
2
c
1
.
(A)
\text{(A)}
(A)
Q
Q
Q
is sufficient and necessary condition of
P
P
P
.
(B)
\text{(B)}
(B)
Q
Q
Q
is sufficient but unnecessary condition of
P
P
P
.
(C)
\text{(C)}
(C)
Q
Q
Q
is insufficient but necessary condition of
P
P
P
.
(D)
\text{(D)}
(D)
Q
Q
Q
is insufficient and unnecessary condition of
P
P
P
.
3
2
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Geometric Series
For geometric series
(
a
n
)
(a_n)
(
a
n
)
with all items real, if
S
10
=
10
,
S
30
=
70
S_{10}=10,S_{30}=70
S
10
=
10
,
S
30
=
70
, then
S
40
=
S_{40}=
S
40
=
(A)
150
(B)
−
200
(C)
150
or
−
200
(D)
−
50
or
400
\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400
(A)
150
(B)
−
200
(C)
150
or
−
200
(D)
−
50
or
400
Note:
S
n
=
∑
i
=
1
n
a
i
S_n=\sum_{i=1}^{n}a_i
S
n
=
∑
i
=
1
n
a
i
.
Number Theory
For positive integers
a
,
n
a,n
a
,
n
, define
F
n
(
a
)
=
q
+
r
F_n(a)=q+r
F
n
(
a
)
=
q
+
r
, where
a
=
q
n
+
r
a=qn+r
a
=
q
n
+
r
(
q
,
r
q,r
q
,
r
are nonnegative integers,
0
≤
q
<
n
0\leq q<n
0
≤
q
<
n
). Find the largest integer
A
A
A
, there are positive integers
n
1
,
n
2
,
n
3
,
n
4
,
n
5
,
n
6
n_1,n_2,n_3,n_4,n_5,n_6
n
1
,
n
2
,
n
3
,
n
4
,
n
5
,
n
6
, for all positive integer
a
≤
A
a\leq A
a
≤
A
,
F
n
6
(
F
n
5
(
F
n
4
(
F
n
3
(
F
n
2
(
F
n
1
(
a
)
)
)
)
)
)
=
1
F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1
F
n
6
(
F
n
5
(
F
n
4
(
F
n
3
(
F
n
2
(
F
n
1
(
a
))))))
=
1
.
2
2
Hide problems
Set Problem
Nonempty set
A
=
{
x
∣
2
a
+
1
≤
x
≤
3
a
−
5
}
,
B
=
{
x
∣
3
≤
x
≤
22
}
A=\{x|2a+1\leq x\leq 3a-5\},B=\{x|3\leq x\leq22\}
A
=
{
x
∣2
a
+
1
≤
x
≤
3
a
−
5
}
,
B
=
{
x
∣3
≤
x
≤
22
}
, then the range value of
a
a
a
such that
A
⊆
A
∩
B
A\subseteq A\cap B
A
⊆
A
∩
B
is
(A)
{
a
∣
1
≤
a
≤
9
}
(B)
{
a
∣
6
≤
a
≤
9
}
(C)
{
a
∣
a
≤
9
}
(D)
∅
\text{(A)}\{a|1\leq a \leq9\}\qquad\text{(B)}\{a|6\leq a \leq9\}\qquad\text{(C)}\{a|a \leq9\}\qquad\text{(D)}\varnothing
(A)
{
a
∣1
≤
a
≤
9
}
(B)
{
a
∣6
≤
a
≤
9
}
(C)
{
a
∣
a
≤
9
}
(D)
∅
Inequality
Let
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
are real numbers in
[
1
,
2
]
[1,2]
[
1
,
2
]
. If
∑
i
=
1
n
a
i
2
=
∑
i
=
1
n
b
i
2
\sum_{i=1}^{n}a_i^2=\sum_{i=1}^{n}b_i^2
∑
i
=
1
n
a
i
2
=
∑
i
=
1
n
b
i
2
, prove that
∑
i
=
1
n
a
i
3
b
i
≤
17
10
∑
i
=
1
n
a
i
2
.
\sum_{i=1}^{n}\frac{a_i^3}{b_i}\leq\frac{17}{10}\sum_{i=1}^{n}a_i^2.
i
=
1
∑
n
b
i
a
i
3
≤
10
17
i
=
1
∑
n
a
i
2
.
Find when the equality holds.
1
2
Hide problems
Logarithm
If
a
>
1
,
b
>
1
,
lg
(
a
+
b
)
=
lg
a
+
lg
b
a>1,b>1,\lg(a+b)=\lg a+\lg b
a
>
1
,
b
>
1
,
l
g
(
a
+
b
)
=
l
g
a
+
l
g
b
, then the value of
lg
(
a
−
1
)
+
lg
(
b
−
1
)
\lg(a-1)+\lg(b-1)
l
g
(
a
−
1
)
+
l
g
(
b
−
1
)
is
(A)
lg
2
(B)
1
(C)
0
(D)
\text{(A)}\lg2\qquad\text{(B)}1\qquad\text{(C)}0\qquad\text{(D)}
(A)
l
g
2
(B)
1
(C)
0
(D)
not sure
Geometry
Circumcenter and incentre of
△
A
B
C
\triangle ABC
△
A
BC
are
O
,
I
O,I
O
,
I
.
A
D
AD
A
D
is the height on side
B
C
BC
BC
. If
I
I
I
is on line
O
C
OC
OC
, prove that the radius of circumcircle and escribed circle (in \angle BAC) are equal.