MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
2000 National High School Mathematics League
2000 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
Hide problems
Analytic Geometry
C
0
:
x
2
+
y
2
=
1
,
C
1
:
x
2
a
2
+
y
2
b
2
(
a
>
b
>
0
)
C_0:x^2+y^2=1,C_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}(a>b>0)
C
0
:
x
2
+
y
2
=
1
,
C
1
:
a
2
x
2
+
b
2
y
2
(
a
>
b
>
0
)
. Find all
(
a
,
b
)
(a,b)
(
a
,
b
)
such that for any point
P
P
P
on
C
1
C_1
C
1
, we can find a parallelogram with an apex
P
P
P
, and it is externally tangent to
C
0
C_0
C
0
, inscribed to
C
1
C_1
C
1
.
14
1
Hide problems
A Function
Function
f
(
x
)
=
−
1
2
x
2
+
13
2
f(x)=-\frac{1}{2}x^2+\frac{13}{2}
f
(
x
)
=
−
2
1
x
2
+
2
13
. If the minumum and maximum value of
f
(
x
)
f(x)
f
(
x
)
are
2
a
2a
2
a
and
2
b
2b
2
b
respectively on
[
a
,
b
]
[a,b]
[
a
,
b
]
. Find
a
,
b
a,b
a
,
b
.
13
1
Hide problems
A Lot of Numbers
Let
S
n
=
1
+
2
+
⋯
+
n
S_n=1+2+\cdots+n
S
n
=
1
+
2
+
⋯
+
n
for
n
∈
N
n\in\mathbb{N}
n
∈
N
, find the maximum value of
f
(
n
)
=
S
n
(
n
+
32
)
S
n
+
1
f(n)=\frac{S_n}{(n+32)S_{n+1}}
f
(
n
)
=
(
n
+
32
)
S
n
+
1
S
n
.
12
1
Hide problems
Count the Number
If
(
1
)
a
,
b
,
c
,
d
∈
{
1
,
2
,
3
,
4
}
;
(
2
)
a
≠
b
,
b
≠
c
,
c
≠
d
,
d
≠
a
;
(
3
)
a
=
min
{
a
,
b
,
c
,
d
}
(1)a,b,c,d\in\{1,2,3,4\};(2)a\neq b,b\neq c,c\neq d,d\neq a;(3)a=\min\{a,b,c,d\}
(
1
)
a
,
b
,
c
,
d
∈
{
1
,
2
,
3
,
4
}
;
(
2
)
a
=
b
,
b
=
c
,
c
=
d
,
d
=
a
;
(
3
)
a
=
min
{
a
,
b
,
c
,
d
}
, then the number of different 4-digit-number
a
b
c
d
‾
\overline{abcd}
ab
c
d
is________.
11
1
Hide problems
3D Geometry
A sphere is tangent to six edges of a regular tetrahedron. If the length of each edge is
a
a
a
, then the volume of the sphere is________.
10
1
Hide problems
Ellipse Problem
In ellipse
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
a
2
x
2
+
b
2
y
2
=
1
,
F
F
F
is its left focal point,
A
A
A
is its right vertex,
B
B
B
is its upper vertex. If the eccentricity of the ellipse is
5
−
1
2
\frac{\sqrt5-1}{2}
2
5
−
1
, then
∠
A
B
F
=
\angle ABF=
∠
A
BF
=
________.
9
1
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Logarithm
If
a
+
log
2
3
,
a
+
log
4
3
,
a
+
log
8
3
a+\log_2 3,a+\log_4 3,a+\log_8 3
a
+
lo
g
2
3
,
a
+
lo
g
4
3
,
a
+
lo
g
8
3
are a geometric series, then the common ratio is________.
8
1
Hide problems
Binomial Theorem
Define
a
n
a_n
a
n
: the coefficient of then item
x
x
x
in
(
3
−
x
)
n
(3-\sqrt{x})^n
(
3
−
x
)
n
, where
n
n
n
is a positive integer. Then
lim
n
→
∞
(
3
2
a
2
+
3
3
a
3
+
⋯
+
3
n
a
n
)
=
\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=
lim
n
→
∞
(
a
2
3
2
+
a
3
3
3
+
⋯
+
a
n
3
n
)
=
________.
7
1
Hide problems
Calculate
arcsin
(
sin
200
0
∘
)
=
\arcsin(\sin 2000^{\circ})=
arcsin
(
sin
200
0
∘
)
=
________.
6
1
Hide problems
Complex Number
Let
ω
=
cos
π
5
+
i
sin
π
5
\omega=\cos\frac{\pi}{5}+\text{i}\sin\frac{\pi}{5}
ω
=
cos
5
π
+
i
sin
5
π
, which equation has roots
ω
,
ω
3
,
ω
7
,
ω
9
\omega,\omega^3,\omega^7,\omega^9
ω
,
ω
3
,
ω
7
,
ω
9
?
(A)
x
4
+
x
3
+
x
2
+
x
+
1
=
0
(B)
x
4
−
x
3
+
x
2
−
x
+
1
=
0
\text{(A)}x^4+x^3+x^2+x+1=0\qquad\text{(B)}x^4-x^3+x^2-x+1=0
(A)
x
4
+
x
3
+
x
2
+
x
+
1
=
0
(B)
x
4
−
x
3
+
x
2
−
x
+
1
=
0
(C)
x
4
−
x
3
−
x
2
+
x
+
1
=
0
(D)
x
4
+
x
3
+
x
2
−
x
+
1
=
0
\text{(C)}x^4-x^3-x^2+x+1=0\qquad\text{(D)}x^4+x^3+x^2-x+1=0
(C)
x
4
−
x
3
−
x
2
+
x
+
1
=
0
(D)
x
4
+
x
3
+
x
2
−
x
+
1
=
0
5
1
Hide problems
Analytic Geometry
The shortest distance from an integral point to line
y
=
5
3
x
+
4
5
y=\frac{5}{3}x+\frac{4}{5}
y
=
3
5
x
+
5
4
is
(A)
34
170
(B)
34
85
(C)
1
20
(D)
1
30
\text{(A)}\frac{\sqrt{34}}{170}\qquad\text{(B)}\frac{\sqrt{34}}{85}\qquad\text{(C)}\frac{1}{20}\qquad\text{(D)}\frac{1}{30}
(A)
170
34
(B)
85
34
(C)
20
1
(D)
30
1
4
1
Hide problems
A Lot of Numbers
Give positive numbers
p
,
q
,
a
,
b
,
c
p,q,a,b,c
p
,
q
,
a
,
b
,
c
, if
p
,
a
,
q
p,a,q
p
,
a
,
q
is a geometric series,
p
,
b
,
c
,
q
p,b,c,q
p
,
b
,
c
,
q
is an arithmetic sequence. Then, wich is true about the equation
b
x
2
−
a
x
+
c
=
0
bx^2-ax+c=0
b
x
2
−
a
x
+
c
=
0
?
(A)
\text{(A)}
(A)
It has no real roots.
(B)
\text{(B)}
(B)
It has two equal real roots.
(C)
\text{(C)}
(C)
It has two different real roots, and their product is positive.
(D)
\text{(D)}
(D)
It has two different real roots, and their product is negative.
3
2
Hide problems
Hyperbola
A
(
−
1
,
1
)
A(-1,1)
A
(
−
1
,
1
)
,
B
,
C
B,C
B
,
C
are points on hyperbola
x
2
−
y
2
=
1
x^2-y^2=1
x
2
−
y
2
=
1
. If
△
A
B
C
\triangle ABC
△
A
BC
is a regular triangle, then the area of
△
A
B
C
\triangle ABC
△
A
BC
is
(A)
3
3
(B)
3
3
2
(C)
3
3
(D)
6
3
\text{(A)}\frac{\sqrt3}{3}\qquad\text{(B)}\frac{3\sqrt3}{2}\qquad\text{(C)}3\sqrt3\qquad\text{(D)}6\sqrt3\qquad
(A)
3
3
(B)
2
3
3
(C)
3
3
(D)
6
3
Call One Another
There are
n
n
n
people, any two of them have called each other at most once. In any group of
n
−
2
n-2
n
−
2
of them, anyone of the group has called with other people in this group for
3
k
3^k
3
k
times, where
k
k
k
is a non-negative integer (the value of
k
k
k
is fixed). Find all possible integers
n
n
n
.
2
2
Hide problems
Trigonometry
If
sin
α
>
0
,
cos
α
<
0
,
sin
α
3
>
cos
α
3
\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}
sin
α
>
0
,
cos
α
<
0
,
sin
3
α
>
cos
3
α
, then the range value of
α
3
\frac{\alpha}{3}
3
α
is
(A)
(
2
k
π
+
π
6
,
2
k
π
+
π
3
)
,
k
∈
Z
\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}
(A)
(
2
kπ
+
6
π
,
2
kπ
+
3
π
)
,
k
∈
Z
(B)
(
2
k
π
3
+
π
6
,
2
k
π
3
+
π
3
)
,
k
∈
Z
\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}
(B)
(
3
2
kπ
+
6
π
,
3
2
kπ
+
3
π
)
,
k
∈
Z
(C)
(
2
k
π
+
5
π
6
,
2
k
π
+
π
)
,
k
∈
Z
\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}
(C)
(
2
kπ
+
6
5
π
,
2
kπ
+
π
)
,
k
∈
Z
(D)
(
2
k
π
+
π
4
,
2
k
π
+
π
3
)
∪
(
2
k
π
+
5
π
6
,
2
k
π
+
π
)
,
k
∈
Z
\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}
(D)
(
2
kπ
+
4
π
,
2
kπ
+
3
π
)
∪
(
2
kπ
+
6
5
π
,
2
kπ
+
π
)
,
k
∈
Z
Number Theory
Two sequences
(
a
n
)
(a_n)
(
a
n
)
and
(
b
n
)
(b_n)
(
b
n
)
satisfy that
a
0
=
1
,
a
1
=
4
,
a
2
=
49
a_0=1,a_1=4,a_2=49
a
0
=
1
,
a
1
=
4
,
a
2
=
49
, and
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
b
n
+
1
=
8
a
n
+
7
b
n
−
4
\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases}
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
b
n
+
1
=
8
a
n
+
7
b
n
−
4
for
n
=
0
,
1
,
2
,
⋯
,
n=0,1,2,\cdots,
n
=
0
,
1
,
2
,
⋯
,
. Prove that
a
n
a_n
a
n
is a perfect square for
n
=
0
,
1
,
2
,
⋯
,
n=0,1,2,\cdots,
n
=
0
,
1
,
2
,
⋯
,
.
1
2
Hide problems
Two Sets
If
A
=
{
x
∣
x
−
2
≤
0
}
,
B
=
{
x
∣
1
0
x
2
−
2
=
1
0
x
}
A=\{x|\sqrt{x-2}\leq0\},B=\{x|10^{x^2-2}=10^{x}\}
A
=
{
x
∣
x
−
2
≤
0
}
,
B
=
{
x
∣1
0
x
2
−
2
=
1
0
x
}
, then
A
∩
(
R
\
B
)
A\cap(\mathbb{R}\backslash B)
A
∩
(
R
\
B
)
is
(A)
{
2
}
(B)
{
−
1
}
(C)
{
x
∣
x
≤
2
}
(D)
∅
\text{(A)}\{2\}\qquad\text{(B)}\{-1\}\qquad\text{(C)}\{x|x\leq2\}\qquad\text{(D)}\varnothing
(A)
{
2
}
(B)
{
−
1
}
(C)
{
x
∣
x
≤
2
}
(D)
∅
Area in Geometry
In acute triangle
A
B
C
ABC
A
BC
,
D
,
E
D,E
D
,
E
are two points on side
B
C
BC
BC
, satisfying that
∠
B
A
E
=
∠
C
A
F
\angle BAE=\angle CAF
∠
B
A
E
=
∠
C
A
F
.
F
M
⊥
A
B
,
E
N
⊥
A
C
FM\perp AB,EN\perp AC
FM
⊥
A
B
,
EN
⊥
A
C
(
M
,
N
M,N
M
,
N
are foot points).
A
E
AE
A
E
intersects the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
D
D
D
. Prove that the area of
△
A
B
C
\triangle ABC
△
A
BC
and quadrilateral
A
M
D
N
AMDN
A
M
D
N
are equal.