MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1999 National High School Mathematics League
1999 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Arithmetic Sequence
Given positive integer
n
n
n
and positive number
M
M
M
. For all arithmetic squence
a
1
,
a
2
,
⋯
,
a_1,a_2,\cdots,
a
1
,
a
2
,
⋯
,
that
a
1
2
+
a
n
+
1
2
≤
M
a_1^2+a_{n+1}^2\leq M
a
1
2
+
a
n
+
1
2
≤
M
, find the maximum value of
S
=
a
n
+
1
+
a
n
+
2
+
⋯
,
a
2
n
+
1
S=a_{n+1}+a_{n+2}+\cdots,a_{2n+1}
S
=
a
n
+
1
+
a
n
+
2
+
⋯
,
a
2
n
+
1
.
14
1
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Ellipse Problem
Given
A
(
−
2
,
2
)
A(-2,2)
A
(
−
2
,
2
)
, and
B
B
B
is a moving point on ellipse
x
2
25
+
y
2
16
=
1
\frac{x^2}{25}+\frac{y^2}{16}=1
25
x
2
+
16
y
2
=
1
.
F
F
F
is the left focal point of the ellipse, find the coordinate of
B
B
B
when
∣
A
B
∣
+
5
3
∣
B
F
∣
|AB|+\frac{5}{3}|BF|
∣
A
B
∣
+
3
5
∣
BF
∣
takes its minumum value.
13
1
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Inequality
If
x
2
cos
θ
−
x
(
1
−
x
)
+
(
1
−
x
)
2
sin
θ
>
0
x^2\cos\theta-x(1-x)+(1-x)^2\sin\theta>0
x
2
cos
θ
−
x
(
1
−
x
)
+
(
1
−
x
)
2
sin
θ
>
0
for all
x
∈
[
0
,
1
]
x\in[0,1]
x
∈
[
0
,
1
]
, find the range value of
θ
\theta
θ
.
12
1
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Triangular Pyramid
The bottom surface of triangular pyramid
S
−
A
B
C
S-ABC
S
−
A
BC
is a regular triangle. Projection of
A
A
A
on plane
S
B
C
SBC
SBC
is
H
H
H
, which is the orthocenter of
△
S
B
C
\triangle SBC
△
SBC
. If
H
−
A
B
−
C
=
3
0
∘
,
S
A
=
2
3
H-AB-C=30^{\circ},SA=2\sqrt3
H
−
A
B
−
C
=
3
0
∘
,
S
A
=
2
3
, then the volume of
S
−
A
B
C
S-ABC
S
−
A
BC
is________.
11
1
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The Number of Lines
Line
l
:
a
x
+
b
y
+
c
=
0
l:ax+by+c=0
l
:
a
x
+
b
y
+
c
=
0
, where
a
,
b
,
c
∈
{
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
}
a,b,c\in\{-3,-2,-1,0,1,2,3\}
a
,
b
,
c
∈
{
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
}
, and
a
,
b
,
c
a,b,c
a
,
b
,
c
are different. If the bank angle of
l
l
l
is an acute angle, then the number of such lines is________.
10
1
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Hyperbola
P
P
P
is a point on hyperbola
x
2
16
−
y
2
9
=
1
\frac{x^2}{16}-\frac{y^2}{9}=1
16
x
2
−
9
y
2
=
1
, if the distance from
P
P
P
to right directrix is the arithmetic mean of the distance from
P
P
P
to two focal points, then the
x
x
x
-axis of
P
P
P
is________.
9
1
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A Triangle
In
△
A
B
C
\triangle ABC
△
A
BC
, if
9
a
2
+
9
b
2
−
19
c
2
=
0
9a^2+9b^2-19c^2=0
9
a
2
+
9
b
2
−
19
c
2
=
0
, then
cot
C
cot
A
+
cot
B
=
\frac{\cot C}{\cot A+\cot B}=
c
o
t
A
+
c
o
t
B
c
o
t
C
=
________.
8
1
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Complex Number
If
θ
=
arctan
5
12
\theta=\arctan \frac{5}{12}
θ
=
arctan
12
5
,
z
=
cos
2
θ
+
i
sin
2
θ
239
+
i
z=\frac{\cos 2\theta+\text{i}\sin2\theta}{239+\text{i}}
z
=
239
+
i
c
o
s
2
θ
+
i
s
i
n
2
θ
, then
arg
z
=
\arg z=
ar
g
z
=
________.
7
1
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Number Theory
Positive integer
n
n
n
is not larger than
2000
2000
2000
, and
n
n
n
is equal to the sum of no less than sixty adjacent positive integers. Then number of such numbers is________.
6
1
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Analytic Geometry
Points
A
(
1
,
2
)
A(1,2)
A
(
1
,
2
)
, a line that passes
(
5
,
−
2
)
(5,-2)
(
5
,
−
2
)
intersects the parabola
y
2
=
4
x
y^2=4x
y
2
=
4
x
at two points
B
,
C
B,C
B
,
C
. Then,
△
A
B
C
\triangle ABC
△
A
BC
is
(A)
\text{(A)}
(A)
an acute triangle
(B)
\text{(B)}
(B)
an obtuse triangle
(C)
\text{(C)}
(C)
a right triangle
(D)
\text{(D)}
(D)
not sure
5
1
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A Ping-pong Game
In a ping-pong game, it was planned to have a competition between any two players. But three players quit the game after having 2 competitions. In the end, the number of competitions played is 50. So the number of competitions between the three players is
(A)
0
(B)
1
(C)
2
(D)
3
\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3
(A)
0
(B)
1
(C)
2
(D)
3
4
1
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Two Statements
Statement 1: Line
a
∈
α
a\in\alpha
a
∈
α
, line
b
∈
β
b\in\beta
b
∈
β
, and
a
,
b
a,b
a
,
b
are skew lines. If
c
=
α
∩
β
c=\alpha\cap\beta
c
=
α
∩
β
, then
c
c
c
intersects at most one of
a
,
b
a,b
a
,
b
. Statement 2: It's impossible to find infintely many lines, any two of them are skew lines.
(A)
\text{(A)}
(A)
Statement 1 is true, Statement 2 is false.
(B)
\text{(B)}
(B)
Statement 2 is true, Statement 1 is false.
(C)
\text{(C)}
(C)
Both are true.
(D)
\text{(D)}
(D)
Neither is true.
3
2
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Logarithm
If
(
log
2
3
)
x
−
(
log
5
3
)
x
≥
(
log
2
3
)
−
y
−
(
log
5
3
)
−
y
(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}
(
lo
g
2
3
)
x
−
(
lo
g
5
3
)
x
≥
(
lo
g
2
3
)
−
y
−
(
lo
g
5
3
)
−
y
, then
(A)
x
−
y
≥
0
(B)
x
+
y
≥
0
(C)
x
−
y
≤
0
(D)
x
+
y
≤
0
\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0
(A)
x
−
y
≥
0
(B)
x
+
y
≥
0
(C)
x
−
y
≤
0
(D)
x
+
y
≤
0
Counter Balance
n
n
n
is a given positive integer, such that it’s possible to weigh out the mass of any product weighing
1
,
2
,
3
,
⋯
,
n
g
1,2,3,\cdots ,n\text{g}
1
,
2
,
3
,
⋯
,
n
g
with a counter balance and
k
k
k
counterweights, whose weights are positive integers. (a) Find
f
(
n
)
f(n)
f
(
n
)
: the minumum value of
k
k
k
. (b) Find all possible number of
n
,
n,
n
,
such that the mass of
f
(
n
)
f(n)
f
(
n
)
counterweights is uniquely determined.
2
2
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Integral Points
The number of intengral points
(
x
,
y
)
(x,y)
(
x
,
y
)
that fit
(
∣
x
∣
−
1
)
2
+
(
∣
y
∣
−
1
)
2
<
2
(|x|-1)^2+(|y|-1)^2<2
(
∣
x
∣
−
1
)
2
+
(
∣
y
∣
−
1
)
2
<
2
is
(A)
16
(B)
17
(C)
18
(D)
25
\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25
(A)
16
(B)
17
(C)
18
(D)
25
Complex Number
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers,
z
1
,
z
2
,
z
3
z_{1},z_{2},z_{3}
z
1
,
z
2
,
z
3
be complex numbers such that
{
∣
z
1
∣
=
∣
z
2
∣
=
∣
z
3
∣
=
1
z
1
z
2
+
z
2
z
3
+
z
3
z
1
=
1
\begin{cases} \displaystyle|z_1|=|z_2|=|z_3|=1\\ \displaystyle\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1\\ \end{cases}
⎩
⎨
⎧
∣
z
1
∣
=
∣
z
2
∣
=
∣
z
3
∣
=
1
z
2
z
1
+
z
3
z
2
+
z
1
z
3
=
1
Find
∣
a
z
1
+
b
z
2
+
c
z
3
∣
|az_{1}+bz_{2}+cz_{3}|
∣
a
z
1
+
b
z
2
+
c
z
3
∣
.
1
2
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Geometric Series
Give a geometric series
(
a
n
)
(a_n)
(
a
n
)
with common ratio of
q
q
q
, let
b
1
=
a
1
+
a
2
+
a
3
,
b
2
=
a
4
+
a
5
+
a
6
,
⋯
,
b
n
=
a
3
n
+
a
3
n
+
1
+
a
3
n
+
2
b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}
b
1
=
a
1
+
a
2
+
a
3
,
b
2
=
a
4
+
a
5
+
a
6
,
⋯
,
b
n
=
a
3
n
+
a
3
n
+
1
+
a
3
n
+
2
, then sequence
(
b
n
)
(b_n)
(
b
n
)
(A)
\text{(A)}
(A)
is an arithmetic sequence
(B)
\text{(B)}
(B)
is a geometric series with common ratio of
q
q
q
(C)
\text{(C)}
(C)
is a geometric series with common ratio of
q
3
q^3
q
3
(D)
\text{(D)}
(D)
is neither an arithmetic sequence nor a geometric series
Easy Geometry
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
∠
B
A
C
=
∠
C
A
D
\angle BAC=\angle CAD
∠
B
A
C
=
∠
C
A
D
.
E
E
E
lies on segment
C
D
CD
C
D
,
B
E
BE
BE
and
A
C
AC
A
C
intersect at
F
,
F,
F
,
D
F
DF
D
F
and
B
C
BC
BC
intersect at
G
.
G.
G
.
Prove that
∠
G
A
C
=
∠
E
A
C
\angle GAC=\angle EAC
∠
G
A
C
=
∠
E
A
C
.