MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1997 National High School Mathematics League
1997 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
Hide problems
Complex Numbers
a
1
,
a
2
,
a
3
,
a
4
,
a
5
a_1,a_2,a_3,a_4,a_5
a
1
,
a
2
,
a
3
,
a
4
,
a
5
are non-zero complex numbers, satisfying:
{
a
2
a
1
=
a
3
a
2
=
a
4
a
3
=
a
5
a
4
a
1
+
a
2
+
a
3
+
a
4
+
a
5
=
4
(
1
a
1
+
1
a
2
+
1
a
3
+
1
a
4
+
1
a
5
)
=
S
\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases}
⎩
⎨
⎧
a
1
a
2
=
a
2
a
3
=
a
3
a
4
=
a
4
a
5
a
1
+
a
2
+
a
3
+
a
4
+
a
5
=
4
(
a
1
1
+
a
2
1
+
a
3
1
+
a
4
1
+
a
5
1
)
=
S
Where
S
S
S
is a real number that
∣
S
∣
≤
2
|S|\leq2
∣
S
∣
≤
2
Prove that points that
a
1
,
a
2
,
a
3
,
a
4
,
a
5
a_1,a_2,a_3,a_4,a_5
a
1
,
a
2
,
a
3
,
a
4
,
a
5
refers to in the complex plane are concyclic.
14
1
Hide problems
Hyperbola
Two branches of the hyperbola
x
y
=
1
xy=1
x
y
=
1
are
C
1
,
C
2
C_1,C_2
C
1
,
C
2
(
C
1
C_1
C
1
in Quadrant I,
C
2
C_2
C
2
in Quadrant III). Three apexes of regular triangle
P
Q
R
PQR
PQR
are on the hyperbola. (a)
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
cannot be on the same branch. (b)
P
(
−
1
,
−
1
)
P(-1,-1)
P
(
−
1
,
−
1
)
is a point on
C
2
C_2
C
2
, if
Q
,
R
Q,R
Q
,
R
are on
C
1
C_1
C
1
, find their coordinates.
13
1
Hide problems
Trigonometry
x
≥
y
≥
z
≥
π
12
,
x
+
y
+
z
=
π
2
x\geq y\geq z\geq \frac{\pi}{12},x+y+z=\frac{\pi}{2}
x
≥
y
≥
z
≥
12
π
,
x
+
y
+
z
=
2
π
, find the maximum and minumum value of
cos
x
sin
y
cos
z
\cos x\sin y\cos z
cos
x
sin
y
cos
z
.
12
1
Hide problems
Logarithm
Let
a
=
lg
z
+
lg
[
x
(
y
z
)
−
1
+
1
]
,
b
=
lg
x
−
1
+
lg
(
x
y
z
+
1
)
,
c
=
lg
y
+
lg
[
(
x
y
z
)
−
1
+
1
]
a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]
a
=
l
g
z
+
l
g
[
x
(
yz
)
−
1
+
1
]
,
b
=
l
g
x
−
1
+
l
g
(
x
yz
+
1
)
,
c
=
l
g
y
+
l
g
[
(
x
yz
)
−
1
+
1
]
, if
M
=
max
{
a
,
b
,
c
}
M=\max\{a,b,c\}
M
=
max
{
a
,
b
,
c
}
, then the minumum value of
M
M
M
is________.
11
1
Hide problems
Jump Around the Regular Hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a regular hexagon. A frog sarts jumping at
A
A
A
, each time it can jump to one of the two adjacent points. If the frog jump to
D
D
D
in no more than five times, it stops. After five jumpings, if the frog hasn't jumped to
D
D
D
yet, it will stop as well. Then the number of different ways to jump is________.
10
1
Hide problems
Triangular Pyramid
Bottom surface of triangular pyramid
S
−
A
B
C
S-ABC
S
−
A
BC
is an isosceles right triangle (hypotenuse is
A
B
AB
A
B
).
S
A
=
S
B
=
S
C
=
A
B
=
2
SA=SB=SC=AB=2
S
A
=
SB
=
SC
=
A
B
=
2
, and
S
,
A
,
B
,
C
S,A,B,C
S
,
A
,
B
,
C
are on a sphere with center of
O
O
O
. The distance of
O
O
O
to plane
A
B
C
ABC
A
BC
is________.
9
1
Hide problems
Complex Number
z
z
z
is a complex number that
∣
2
z
+
1
z
∣
=
1
\left|2z+\frac{1}{z}\right|=1
2
z
+
z
1
=
1
, then the range value of
arg
(
z
)
\arg(z)
ar
g
(
z
)
is________.
8
1
Hide problems
Hyperbola
Line
l
l
l
that passes right focal point of hyperbola
x
2
−
y
2
2
=
1
x^2-\frac{y^2}{2}=1
x
2
−
2
y
2
=
1
intersects the hyperbola at
A
,
B
A,B
A
,
B
. The number of line
l
l
l
that
∣
A
B
∣
=
λ
|AB|=\lambda
∣
A
B
∣
=
λ
is 3, then
λ
=
\lambda=
λ
=
________.
7
1
Hide problems
Two Numbers
Real numbers
x
,
y
x,y
x
,
y
satisfy that
{
(
x
−
1
)
3
+
1997
(
x
−
1
)
=
−
1
(
y
−
1
)
3
+
1997
(
y
−
1
)
=
1
\begin{cases} (x-1)^3+1997(x-1)=-1\\ (y-1)^3+1997(y-1)=1 \end{cases}
{
(
x
−
1
)
3
+
1997
(
x
−
1
)
=
−
1
(
y
−
1
)
3
+
1997
(
y
−
1
)
=
1
, then
x
+
y
=
x+y=
x
+
y
=
________.
6
1
Hide problems
The Number of Lines
In the space, three lines
a
,
b
,
c
a,b,c
a
,
b
,
c
that any two in them are skew lines. Then the number of lines that intersect all of
a
,
b
,
c
a,b,c
a
,
b
,
c
is
(A)
0
(B)
1
(C)more than one, but finitely many
(D)infinitely many
\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}\text{more than one, but finitely many}\qquad\text{(D)} \text{infinitely many}
(A)
0
(B)
1
(C)
more than one, but finitely many
(D)
infinitely many
5
1
Hide problems
Anti-trigonometry Function
Let
f
(
x
)
=
x
2
−
π
x
f(x)=x^2-\pi x
f
(
x
)
=
x
2
−
π
x
,
α
=
arcsin
1
3
,
β
=
arctan
5
4
,
γ
=
arccos
(
−
1
3
)
,
δ
=
arccot
(
−
5
4
)
\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)
α
=
arcsin
3
1
,
β
=
arctan
4
5
,
γ
=
arccos
(
−
3
1
)
,
δ
=
arccot
(
−
4
5
)
(A)
f
(
α
)
>
f
(
β
)
>
f
(
δ
)
>
f
(
γ
)
\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)
(A)
f
(
α
)
>
f
(
β
)
>
f
(
δ
)
>
f
(
γ
)
(B)
f
(
α
)
>
f
(
δ
)
>
f
(
β
)
>
f
(
γ
)
\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)
(B)
f
(
α
)
>
f
(
δ
)
>
f
(
β
)
>
f
(
γ
)
(C)
f
(
δ
)
>
f
(
α
)
>
f
(
β
)
>
f
(
γ
)
\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)
(C)
f
(
δ
)
>
f
(
α
)
>
f
(
β
)
>
f
(
γ
)
(D)
f
(
δ
)
>
f
(
α
)
>
f
(
γ
)
>
f
(
β
)
\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)
(D)
f
(
δ
)
>
f
(
α
)
>
f
(
γ
)
>
f
(
β
)
4
1
Hide problems
Rectangular Coordinate System
In rectangular coordinate system, if
m
(
x
2
+
y
2
+
2
y
+
1
)
=
(
x
−
2
y
+
3
)
2
m(x^2+y^2+2y+1)=(x-2y+3)^2
m
(
x
2
+
y
2
+
2
y
+
1
)
=
(
x
−
2
y
+
3
)
2
refers to an ellipse, then the range value of
m
m
m
is
(A)
(
0
,
1
)
(B)
(
1
,
+
∞
)
(C)
(
0
,
5
)
(D)
(
5
,
+
∞
)
\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)
(A)
(
0
,
1
)
(B)
(
1
,
+
∞
)
(C)
(
0
,
5
)
(D)
(
5
,
+
∞
)
3
2
Hide problems
Arithmetic Sequence
The first item and common difference of an arithmetic sequence are nonnegative intengers. The number of items is not less than
3
3
3
, and the sum of all items is
9
7
2
97^2
9
7
2
. Then the number of such sequences is
(A)
2
(B)
3
(C)
4
(D)
5
\text{(A)}2\qquad\text{(B)}3\qquad\text{(C)}4\qquad\text{(D)}5
(A)
2
(B)
3
(C)
4
(D)
5
A Large Table
In a
100
×
25
100\times25
100
×
25
rectangle table, fill in a positive real number in each blank. Let the number in the
i
i
i
th line, the
j
j
j
th column be
x
i
,
j
(
i
=
1
,
2
,
⋯
,
100
,
j
=
1
,
2
,
⋯
,
25
)
x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)
x
i
,
j
(
i
=
1
,
2
,
⋯
,
100
,
j
=
1
,
2
,
⋯
,
25
)
(shown in Fig.1 ). Then, we rearrange the numbers in each column:
x
1
,
j
′
≥
x
2
,
j
′
≥
⋯
≥
x
100
,
j
′
(
j
=
1
,
2
,
⋯
,
25
)
x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)
x
1
,
j
′
≥
x
2
,
j
′
≥
⋯
≥
x
100
,
j
′
(
j
=
1
,
2
,
⋯
,
25
)
(shown in Fig.2 ). Find the minumum value of
k
k
k
, satisfying: As long as
∑
j
=
1
25
x
i
,
j
≤
1
\sum_{j=1}^{25}x_{i,j}\leq1
∑
j
=
1
25
x
i
,
j
≤
1
for numbers in Fig.1 (
i
=
1
,
2
,
⋯
,
100
i=1,2,\cdots,100
i
=
1
,
2
,
⋯
,
100
), then
∑
j
=
1
25
x
i
,
j
′
≤
1
\sum_{j=1}^{25}x'_{i,j}\leq1
∑
j
=
1
25
x
i
,
j
′
≤
1
for
i
≥
k
i\geq k
i
≥
k
in Fig.2.
Fig.1
\\ \begin{tabular}{|c|c|c|c|} \hline $x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\ \hline $x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\ \hline \end{tabular} \qquad
Fig.2
\\ \begin{tabular}{|c|c|c|c|} \hline $x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\ \hline $x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\ \hline \end{tabular}
2
2
Hide problems
Regular Tetrahedron
In regular tetrahedron
A
B
C
D
ABCD
A
BC
D
,
E
∈
A
B
,
F
∈
C
D
E\in AB,F\in CD
E
∈
A
B
,
F
∈
C
D
, satisfying:
∣
A
E
∣
∣
E
B
∣
=
∣
C
F
∣
∣
F
D
∣
=
λ
(
λ
∈
R
+
)
\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)
∣
EB
∣
∣
A
E
∣
=
∣
F
D
∣
∣
CF
∣
=
λ
(
λ
∈
R
+
)
. Note that
f
(
λ
)
=
α
λ
+
β
λ
f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}
f
(
λ
)
=
α
λ
+
β
λ
, where
α
λ
=
<
E
F
,
A
C
>
,
α
λ
=
<
E
F
,
B
D
>
\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>
α
λ
=<
EF
,
A
C
>
,
α
λ
=<
EF
,
B
D
>
.
(A)
\text{(A)}
(A)
f
(
λ
)
f(\lambda)
f
(
λ
)
increases in
(
0
,
+
∞
)
(0,+\infty)
(
0
,
+
∞
)
(B)
\text{(B)}
(B)
f
(
λ
)
f(\lambda)
f
(
λ
)
decreases in
(
0
,
+
∞
)
(0,+\infty)
(
0
,
+
∞
)
(C)
\text{(C)}
(C)
f
(
λ
)
f(\lambda)
f
(
λ
)
increases in
(
0
,
1
)
(0,1)
(
0
,
1
)
, decreases in
(
1
,
+
∞
)
(1,+\infty)
(
1
,
+
∞
)
(D)
\text{(D)}
(D)
f
(
λ
)
f(\lambda)
f
(
λ
)
is a fixed value in
(
0
,
+
∞
)
(0,+\infty)
(
0
,
+
∞
)
Complex Number
For real numbers
x
0
,
x
1
,
⋯
,
x
n
x_0,x_1,\cdots,x_n
x
0
,
x
1
,
⋯
,
x
n
, there exists real numbers
y
0
,
y
1
,
⋯
,
y
n
y_0,y_1,\cdots,y_n
y
0
,
y
1
,
⋯
,
y
n
, satisfying that
z
0
2
=
z
1
2
+
z
2
2
+
⋯
+
z
n
2
z_0^2=z_1^2+z_2^2+\cdots+z_n^2
z
0
2
=
z
1
2
+
z
2
2
+
⋯
+
z
n
2
, where
z
k
=
x
k
+
i
y
k
(
k
=
0
,
1
,
⋯
,
n
)
z_k=x_k+\text{i}y_{k}(k=0,1,\cdots,n)
z
k
=
x
k
+
i
y
k
(
k
=
0
,
1
,
⋯
,
n
)
. Find all such
(
x
0
,
x
1
,
⋯
,
x
n
)
(x_0,x_1,\cdots,x_n)
(
x
0
,
x
1
,
⋯
,
x
n
)
.
1
2
Hide problems
Sequence
Squence
(
x
n
)
(x_n)
(
x
n
)
satisfies that
x
n
+
1
=
x
n
−
x
n
−
1
(
n
≥
2
)
x_{n+1}=x_n-x_{n-1}(n\geq2)
x
n
+
1
=
x
n
−
x
n
−
1
(
n
≥
2
)
. If
x
1
=
a
,
x
2
=
b
x_1=a,x_2=b
x
1
=
a
,
x
2
=
b
,
S
n
=
x
1
+
x
2
+
⋯
+
x
n
S_n=x_1+x_2+\cdots+x_n
S
n
=
x
1
+
x
2
+
⋯
+
x
n
. Wich one is correct?
(A)
x
100
=
−
a
,
S
100
=
2
b
−
a
\text{(A)}x_{100}=-a,S_{100}=2b-a
(A)
x
100
=
−
a
,
S
100
=
2
b
−
a
(B)
x
100
=
−
b
,
S
100
=
2
b
−
a
\text{(B)}x_{100}=-b,S_{100}=2b-a
(B)
x
100
=
−
b
,
S
100
=
2
b
−
a
(C)
x
100
=
−
a
,
S
100
=
b
−
a
\text{(C)}x_{100}=-a,S_{100}=b-a
(C)
x
100
=
−
a
,
S
100
=
b
−
a
(D)
x
100
=
−
b
,
S
100
=
b
−
a
\text{(D)}x_{100}=-b,S_{100}=b-a
(D)
x
100
=
−
b
,
S
100
=
b
−
a
Geometry
Two circles with different radius
O
1
O_1
O
1
and
O
2
O_2
O
2
are both tangent to a larger circle
O
O
O
, tangent points are
S
,
T
S,T
S
,
T
. Note that intersections of
O
1
O_1
O
1
and
O
2
O_2
O
2
are
M
,
N
M,N
M
,
N
, prove that the sufficient and necessary condition of
O
M
⊥
M
N
OM\perp MN
OM
⊥
MN
is
S
,
N
,
T
S,N,T
S
,
N
,
T
are colinear.