MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1986 National High School Mathematics League
1986 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(10)
10
1
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Max and Min
x
,
y
,
z
x,y,z
x
,
y
,
z
are nonnegative real numbers, and
4
5
x
+
9
y
+
4
z
−
68
×
2
5
x
+
9
y
+
4
z
+
256
=
0
4^{\sqrt{5x+9y+4z}}-68\times2^{\sqrt{5x+9y+4z}}+256=0
4
5
x
+
9
y
+
4
z
−
68
×
2
5
x
+
9
y
+
4
z
+
256
=
0
. Then, the product of the maximum and minimum value of
x
+
y
+
z
x+y+z
x
+
y
+
z
is________.
9
1
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Function Calculation
f
(
x
)
=
4
x
4
x
+
2
f(x)=\frac{4^x}{4^x+2}
f
(
x
)
=
4
x
+
2
4
x
, then
f
(
1
1001
)
+
f
(
2
1001
)
+
⋯
+
f
(
1000
1001
)
=
f(\frac{1}{1001})+f(\frac{2}{1001})+\cdots+f(\frac{1000}{1001})=
f
(
1001
1
)
+
f
(
1001
2
)
+
⋯
+
f
(
1001
1000
)
=
________.
8
1
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Function Iteration
f
(
x
)
=
∣
1
−
2
x
∣
,
x
∈
[
0
,
1
]
f(x)=|1-2x|,x\in[0,1]
f
(
x
)
=
∣1
−
2
x
∣
,
x
∈
[
0
,
1
]
. Then the number of solutions to
f
(
f
(
f
(
x
)
)
)
=
1
2
x
f(f(f(x)))=\frac{1}{2}x
f
(
f
(
f
(
x
)))
=
2
1
x
is________.
7
1
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3D Problem
Inside a circular column with a bottom surface with radius of
6
6
6
, there are two balls with radius of
6
6
6
as well. The distance betwen their centers is
13
13
13
. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve
C
C
C
.
C
C
C
is a ellipse, then the sum of its short axis and long axis is________.
6
1
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Triangle Problem
Area of
△
A
B
C
\triangle ABC
△
A
BC
is
1
4
\frac{1}{4}
4
1
, circumradius of
△
A
B
C
\triangle ABC
△
A
BC
is
1
1
1
. Let
s
=
a
+
b
+
c
,
t
=
1
a
+
1
b
+
1
c
s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}
s
=
a
+
b
+
c
,
t
=
a
1
+
b
1
+
c
1
, then
(A)
s
>
t
(B)
s
=
t
(C)
s
<
t
(D)
s
>
t
\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t
(A)
s
>
t
(B)
s
=
t
(C)
s
<
t
(D)
s
>
t
5
1
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A Point Set and Seven Circles
There is a point set on a plane, and seven circles
C
1
,
C
2
,
⋯
,
C
7
C_1,C_2,\cdots,C_7
C
1
,
C
2
,
⋯
,
C
7
, where
C
7
C_7
C
7
passes exactly 7 points in
M
M
M
,
C
6
C_6
C
6
passes exactly 6 points in
M
M
M
, ...,
C
1
C_1
C
1
passes exactly 1 point in
M
M
M
. Then how many points do set
M
M
M
have at least?
(A)
11
(B)
12
(C)
21
(D)
28
\text{(A)}11\qquad\text{(B)}12\qquad\text{(C)}21\qquad\text{(D)}28
(A)
11
(B)
12
(C)
21
(D)
28
4
1
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3D Problem
None face of a tetrahedron is isosceles triangle. How many kinds of lengths of edges do the tetrahedron have at least?
(A)
3
(B)
4
(C)
5
(D)
6
\text{(A)}3\qquad\text{(B)}4\qquad\text{(C)}5\qquad\text{(D)}6
(A)
3
(B)
4
(C)
5
(D)
6
3
2
Hide problems
Inequality?
For real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, if
a
2
−
b
c
−
8
a
+
7
=
b
2
+
c
2
+
b
c
−
6
a
−
6
=
0
,
a^2-bc-8a+7=b^2+c^2+bc-6a-6=0,
a
2
−
b
c
−
8
a
+
7
=
b
2
+
c
2
+
b
c
−
6
a
−
6
=
0
,
then the range value of
a
a
a
is
(A)
(
−
∞
,
+
∞
)
(B)
(
−
∞
,
1
]
∪
[
9
,
+
∞
)
(C)
(
0
,
7
)
(D)
[
1
,
9
]
\text{(A)}(-\infty,+\infty)\qquad\text{(B)}(-\infty,1]\cup[9,+\infty)\qquad\text{(C)}(0,7)\qquad\text{(D)}[1,9]
(A)
(
−
∞
,
+
∞
)
(B)
(
−
∞
,
1
]
∪
[
9
,
+
∞
)
(C)
(
0
,
7
)
(D)
[
1
,
9
]
Integral Points
In rectangular coordinate system, define that if and only if both
x
x
x
-axis and
y
y
y
-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to
x
x
x
-axis. (2) For any white point
A
A
A
, red point
B
B
B
, black point
C
C
C
, we can find another red point
D
D
D
, such that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
2
2
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Complex Number Problem
Set
M
=
{
z
∈
C
∣
(
z
−
1
)
2
=
∣
z
−
1
∣
2
}
M=\{z\in\mathbb{C}|(z-1)^2=|z-1|^2\}
M
=
{
z
∈
C
∣
(
z
−
1
)
2
=
∣
z
−
1
∣
2
}
, then
(A)
M
=
{
pure imaginary number
}
\text{(A)}M=\{\text{pure imaginary number}\}
(A)
M
=
{
pure imaginary number
}
(B)
M
=
{
real number
}
\text{(B)}M=\{\text{real number}\}
(B)
M
=
{
real number
}
(C)
M
=
{
real number
}
⊂
M
⊂
{
complex number
}
\text{(C)}M=\{\text{real number}\}\subset M\subset\{\text{complex number}\}
(C)
M
=
{
real number
}
⊂
M
⊂
{
complex number
}
(D)
M
=
{
complex number
}
\text{(D)}M=\{\text{complex number}\}
(D)
M
=
{
complex number
}
Area of the Triangle
In acute triangle
A
B
C
ABC
A
BC
,
D
∈
B
C
,
E
∈
C
A
,
F
∈
A
B
D\in BC,E\in CA,F\in AB
D
∈
BC
,
E
∈
C
A
,
F
∈
A
B
. Prove that the necessary and sufficient condition of
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
are heights of
△
A
B
C
\triangle ABC
△
A
BC
is that
S
=
R
2
(
E
F
+
F
D
+
D
E
)
S=\frac{R}{2}(EF+FD+DE)
S
=
2
R
(
EF
+
F
D
+
D
E
)
. Note:
S
S
S
is the area of
△
A
B
C
\triangle ABC
△
A
BC
,
R
R
R
is the circumradius of
△
A
B
C
\triangle ABC
△
A
BC
.
1
2
Hide problems
Anti-trigonometric Function Problem
Let
−
1
<
a
<
0
-1<a<0
−
1
<
a
<
0
,
θ
=
arcsin
a
\theta=\arcsin a
θ
=
arcsin
a
. Then the solution set to the inequality
sin
x
<
a
\sin x<a
sin
x
<
a
is
(A)
{
x
∣
2
n
π
+
θ
<
x
<
(
2
n
+
1
)
π
−
θ
,
n
∈
Z
}
\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}
(A)
{
x
∣2
nπ
+
θ
<
x
<
(
2
n
+
1
)
π
−
θ
,
n
∈
Z
}
(B)
{
x
∣
2
n
π
−
θ
<
x
<
(
2
n
+
1
)
π
+
θ
,
n
∈
Z
}
\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}
(B)
{
x
∣2
nπ
−
θ
<
x
<
(
2
n
+
1
)
π
+
θ
,
n
∈
Z
}
(C)
{
x
∣
(
2
n
−
1
)
π
+
θ
<
x
<
2
n
π
−
θ
,
n
∈
Z
}
\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}
(C)
{
x
∣
(
2
n
−
1
)
π
+
θ
<
x
<
2
nπ
−
θ
,
n
∈
Z
}
(D)
{
x
∣
(
2
n
−
1
)
π
−
θ
<
x
<
2
n
π
+
θ
,
n
∈
Z
}
\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}
(D)
{
x
∣
(
2
n
−
1
)
π
−
θ
<
x
<
2
nπ
+
θ
,
n
∈
Z
}
Linear Polynomial
For real numbers
a
0
,
a
1
,
⋯
,
a
n
(
a
0
≠
a
1
)
a_0,a_1,\cdots,a_n(a_0\neq a_1)
a
0
,
a
1
,
⋯
,
a
n
(
a
0
=
a
1
)
, we have
a
i
−
1
+
a
i
+
1
=
2
a
i
a_{i-1}+a_{i+1}=2a_i
a
i
−
1
+
a
i
+
1
=
2
a
i
for
i
=
1
,
2
,
⋯
,
n
−
1
i=1,2,\cdots,n-1
i
=
1
,
2
,
⋯
,
n
−
1
. Prove that
P
(
x
)
=
a
0
C
n
0
(
1
−
x
)
n
+
a
1
C
n
1
x
(
1
−
x
)
n
−
1
+
⋯
+
a
n
C
n
n
x
n
P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n
P
(
x
)
=
a
0
C
n
0
(
1
−
x
)
n
+
a
1
C
n
1
x
(
1
−
x
)
n
−
1
+
⋯
+
a
n
C
n
n
x
n
is a linear polynomial.