MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1990 National High School Mathematics League
1990 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Pyramid Problem
In pyramid
M
−
A
B
C
D
M-ABCD
M
−
A
BC
D
, bottom surface
A
B
C
D
ABCD
A
BC
D
is a square.
M
A
=
M
C
,
M
A
⊥
A
B
MA=MC,MA\perp AB
M
A
=
MC
,
M
A
⊥
A
B
. If the area of
△
A
M
D
\triangle AMD
△
A
M
D
is
1
1
1
, find the maximum value of radius of sphere that can be put inside the pyramid.
14
1
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Numerical Table
Here are
n
2
n^2
n
2
numbers:
a
11
,
a
12
,
a
13
,
⋯
,
a
1
n
a
21
,
a
22
,
a
23
,
⋯
,
a
2
n
⋯
a
n
1
,
a
n
2
,
a
n
3
,
⋯
,
a
n
n
a_{11},a_{12},a_{13},\cdots,a_{1n}\\ a_{21},a_{22},a_{23},\cdots,a_{2n}\\ \cdots\\ a_{n1},a_{n2},a_{n3},\cdots,a_{nn}
a
11
,
a
12
,
a
13
,
⋯
,
a
1
n
a
21
,
a
22
,
a
23
,
⋯
,
a
2
n
⋯
a
n
1
,
a
n
2
,
a
n
3
,
⋯
,
a
nn
Numbers in each line are arithmetic sequence, numbers in each column are geometric series. If
a
24
=
1
,
a
42
=
1
8
,
a
43
=
3
16
a_{24}=1,a_{42}=\frac{1}{8},a_{43}=\frac{3}{16}
a
24
=
1
,
a
42
=
8
1
,
a
43
=
16
3
, find
a
11
+
a
22
+
⋯
+
a
n
n
a_{11}+a_{22}+\cdots+a_{nn}
a
11
+
a
22
+
⋯
+
a
nn
.
13
1
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Trigonometric Functions
a
,
b
a,b
a
,
b
are positive integers,
a
>
b
a>b
a
>
b
.
sin
θ
=
2
a
b
a
2
+
b
2
(
0
<
θ
<
π
2
)
\sin\theta=\frac{2ab}{a^2+b^2}(0<\theta<\frac{\pi}{2})
sin
θ
=
a
2
+
b
2
2
ab
(
0
<
θ
<
2
π
)
. If
A
n
=
(
a
2
+
b
2
)
sin
n
θ
A_n=(a^2+b^2)\sin n\theta
A
n
=
(
a
2
+
b
2
)
sin
n
θ
, prove that
A
n
A_n
A
n
is an integer for all
n
∈
Z
+
n\in\mathbb{Z}_+
n
∈
Z
+
12
1
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Stand in a Circle
8
8
8
girls and
25
25
25
boys stand in a circle. Between two girls, there are at least two boys. So, we have________ways.
11
1
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Calculate
1
2
1990
(
1
−
3
C
1990
2
+
3
2
C
1990
4
−
3
3
C
1990
6
+
⋯
+
3
994
C
1990
1988
−
3
995
C
1990
1990
)
=
\frac{1}{2^{1990}}(1-3\text{C}_{1990}^2+3^2\text{C}_{1990}^4-3^3\text{C}_{1990}^6+\cdots+3^{994}\text{C}_{1990}^{1988}-3^{995}\text{C}_{1990}^{1990})=
2
1990
1
(
1
−
3
C
1990
2
+
3
2
C
1990
4
−
3
3
C
1990
6
+
⋯
+
3
994
C
1990
1988
−
3
995
C
1990
1990
)
=
________.
10
1
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The Number of Integral Points
Define
f
(
n
)
:
f(n):
f
(
n
)
:
the number of integral points of line segment
O
A
n
OA_n
O
A
n
(
O
O
O
and
A
n
A_n
A
n
not included), where
A
n
(
n
,
n
+
3
)
A_n(n,n+3)
A
n
(
n
,
n
+
3
)
. Then,
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
1990
)
=
f(1)+f(2)+\cdots+f(1990)=
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
1990
)
=
________.
9
1
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Inequality
Let
n
n
n
be a natural number. For all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
,
(
x
2
+
y
2
+
z
2
)
2
≥
n
(
x
4
+
y
4
+
z
4
)
(x^2+y^2+z^2)^2\geq n(x^4+y^4+z^4)
(
x
2
+
y
2
+
z
2
)
2
≥
n
(
x
4
+
y
4
+
z
4
)
, then the minumum value of
n
n
n
is________.
8
1
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Area Swept by Segment
Point
A
(
2
,
0
)
A(2,0)
A
(
2
,
0
)
.
P
(
sin
(
2
t
−
π
3
)
,
cos
(
2
t
−
π
3
)
)
P(\sin(2t-\frac{\pi}{3}),\cos(2t-\frac{\pi}{3}))
P
(
sin
(
2
t
−
3
π
)
,
cos
(
2
t
−
3
π
))
is a moving point. When
t
t
t
changes from
π
12
\frac{\pi}{12}
12
π
to
π
4
\frac{\pi}{4}
4
π
, area swept by segment
A
P
AP
A
P
is________.
7
1
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Inequality
If
n
∈
Z
+
n\in\mathbb{Z_+}
n
∈
Z
+
, positive real numbers
a
+
b
=
2
a+b=2
a
+
b
=
2
, then the minumum value of
1
1
+
a
n
+
1
1
+
b
n
\frac{1}{1+a^n}+\frac{1}{1+b^n}
1
+
a
n
1
+
1
+
b
n
1
is________.
6
1
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A Boring Problem, Please Ignore It...
An ellipse
x
2
a
2
+
y
2
b
2
=
1
(
a
>
b
>
0
)
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)
a
2
x
2
+
b
2
y
2
=
1
(
a
>
b
>
0
)
passes point
(
2
,
1
)
(2,1)
(
2
,
1
)
, then all points
(
x
,
y
)
(x,y)
(
x
,
y
)
on the ellipse that
∣
y
∣
>
1
|y|>1
∣
y
∣
>
1
are (shown as shadow) https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg
5
1
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Complex Number
Two non-zero-complex numbers
x
,
y
x,y
x
,
y
, satisfy that
x
2
+
x
y
+
y
2
=
0
x^2+xy+y^2=0
x
2
+
x
y
+
y
2
=
0
. Then the value of
(
x
x
+
y
)
1990
+
(
y
x
+
y
)
1990
(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}
(
x
+
y
x
)
1990
+
(
x
+
y
y
)
1990
is
(A)
2
−
1989
(B)
−
1
(C)
1
(D)
\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}
(A)
2
−
1989
(B)
−
1
(C)
1
(D)
none above
4
1
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Problem of a Set
The number of points in the set
{
(
x
,
y
)
∣
lg
(
x
3
+
1
3
y
3
+
1
9
)
=
lg
x
+
lg
y
)
}
\{(x,y)|\lg(x^3+\frac{1}{3}y^3+\frac{1}{9})=\lg x+\lg y)\}
{(
x
,
y
)
∣
l
g
(
x
3
+
3
1
y
3
+
9
1
)
=
l
g
x
+
l
g
y
)}
is
(A)
0
(B)
1
(C)
2
(D)
\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}
(A)
0
(B)
1
(C)
2
(D)
more than
2
2
2
3
2
Hide problems
Hyperbola Problem
Left focal point and right focal point of a hyperbola are
F
1
,
F
2
F_1,F_2
F
1
,
F
2
, left focal point and right focal point of a hyperbola are
M
,
N
M,N
M
,
N
. If
P
P
P
is a point on the hyperbola, then the tangent point of inscribed circle of
△
P
F
1
F
2
\triangle PF_1F_2
△
P
F
1
F
2
on
F
1
F
2
F_1F_2
F
1
F
2
is
(A)
\text{(A)}
(A)
a point on segment
M
N
MN
MN
(B)
\text{(B)}
(B)
a point on segment
F
1
M
F_1M
F
1
M
or
F
2
N
F_2N
F
2
N
(C)
\text{(C)}
(C)
point
M
M
M
or
N
N
N
(D)
\text{(D)}
(D)
not sure
Watch a Football Match
There are
n
n
n
schools in a city.
i
i
i
th school dispatches
C
i
(
1
≤
C
i
≤
39
,
1
≤
i
≤
n
)
C_i(1\leq C_i\leq39,1\leq i\leq n)
C
i
(
1
≤
C
i
≤
39
,
1
≤
i
≤
n
)
students to watch a football match. The number of all students
∑
i
=
1
n
C
i
=
1990
\sum_{i=1}^{n}C_{i}=1990
∑
i
=
1
n
C
i
=
1990
. In each line, there are
199
199
199
seats, but students from the same school must sit in the same line. So, how many lines of seats we need to have to make sure all students have a seat.
2
2
Hide problems
Periodic Function
f
(
x
)
f(x)
f
(
x
)
is a periodic even function defined on
R
\mathbb{R}
R
, with period of
2
2
2
. When
x
∈
[
2
,
3
]
x\in[2,3]
x
∈
[
2
,
3
]
,
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
. Then what's
f
(
x
)
f(x)
f
(
x
)
if
x
∈
[
−
2
,
0
]
x\in[-2,0]
x
∈
[
−
2
,
0
]
?
(A)
f
(
x
)
=
x
+
4
(B)
f
(
x
)
=
2
−
x
(C)
f
(
x
)
=
3
−
∣
x
+
1
∣
(D)
f
(
x
)
=
2
+
∣
x
+
1
∣
\text{(A)}f(x)=x+4\qquad\text{(B)}f(x)=2-x\qquad\text{(C)}f(x)=3-|x+1|\qquad\text{(D)}f(x)=2+|x+1|
(A)
f
(
x
)
=
x
+
4
(B)
f
(
x
)
=
2
−
x
(C)
f
(
x
)
=
3
−
∣
x
+
1∣
(D)
f
(
x
)
=
2
+
∣
x
+
1∣
A Set of 100 Elements
E
=
{
1
,
2
,
⋯
,
200
}
,
G
=
{
a
1
,
a
2
,
⋯
,
a
100
}
⊂
E
E=\{1,2,\cdots,200\},G=\{a_1,a_2,\cdots,a_{100}\}\subset E
E
=
{
1
,
2
,
⋯
,
200
}
,
G
=
{
a
1
,
a
2
,
⋯
,
a
100
}
⊂
E
.
G
G
G
satisfies the following: (1)For any
1
≥
i
<
j
≥
100
1\geq i<j\geq100
1
≥
i
<
j
≥
100
, a_i+a_j\neq201. (2)
∑
i
=
1
100
a
i
=
10080
\sum_{i=1}^{100}a_i=10080
∑
i
=
1
100
a
i
=
10080
. Prove that the number of odd numbers in
G
G
G
is a multiple of
4
4
4
, and the sum the square of all numbers in
G
G
G
is fixed.
1
2
Hide problems
Order the Numbers
Let
α
∈
(
π
4
,
π
2
)
\alpha\in(\frac{\pi}{4},\frac{\pi}{2})
α
∈
(
4
π
,
2
π
)
, then the order of
(
cos
α
)
cos
α
,
(
sin
α
)
cos
α
,
(
cos
α
)
sin
α
(\cos\alpha)^{\cos\alpha},(\sin\alpha)^{\cos\alpha},(\cos\alpha)^{\sin\alpha}
(
cos
α
)
c
o
s
α
,
(
sin
α
)
c
o
s
α
,
(
cos
α
)
s
i
n
α
is
(A)
(
cos
α
)
cos
α
<
(
sin
α
)
cos
α
<
(
cos
α
)
sin
α
\text{(A)}(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}
(A)
(
cos
α
)
c
o
s
α
<
(
sin
α
)
c
o
s
α
<
(
cos
α
)
s
i
n
α
(B)
(
cos
α
)
cos
α
<
(
cos
α
)
sin
α
<
(
sin
α
)
cos
α
\text{(B)}(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}<(\sin\alpha)^{\cos\alpha}
(B)
(
cos
α
)
c
o
s
α
<
(
cos
α
)
s
i
n
α
<
(
sin
α
)
c
o
s
α
(C)
(
sin
α
)
cos
α
<
(
cos
α
)
cos
α
<
(
cos
α
)
sin
α
\text{(C)}(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}
(C)
(
sin
α
)
c
o
s
α
<
(
cos
α
)
c
o
s
α
<
(
cos
α
)
s
i
n
α
(D)
(
cos
α
)
sin
α
<
(
cos
α
)
cos
α
<
(
sin
α
)
cos
α
\text{(D)}(\cos\alpha)^{\sin\alpha}<(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}
(D)
(
cos
α
)
s
i
n
α
<
(
cos
α
)
c
o
s
α
<
(
sin
α
)
c
o
s
α
Geometry~
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed on circle
O
O
O
.
A
C
∩
B
D
=
P
AC\cap BD=P
A
C
∩
B
D
=
P
. Circumcenters of
△
A
B
P
,
△
B
C
P
,
△
C
D
P
,
△
D
A
P
\triangle ABP,\triangle BCP,\triangle CDP,\triangle DAP
△
A
BP
,
△
BCP
,
△
C
D
P
,
△
D
A
P
are
O
1
,
O
2
,
O
3
,
O
4
O_1,O_2,O_3,O_4
O
1
,
O
2
,
O
3
,
O
4
. Prove that
O
P
,
O
1
O
3
,
O
2
O
4
OP,O_1O_3,O_2O_4
OP
,
O
1
O
3
,
O
2
O
4
share one point.