MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1993 National High School Mathematics League
1993 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Find the General Formulas
Positive sequence
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
satisfies that
a
n
a
n
−
2
−
a
n
−
1
a
n
−
2
=
2
a
n
−
1
(
n
≥
2
)
,
a
0
=
a
1
=
1
\sqrt{a_na_{n-2}}-\sqrt{a_{n-1}a_{n-2}}=2a_{n-1}(n\geq2),a_0=a_1=1
a
n
a
n
−
2
−
a
n
−
1
a
n
−
2
=
2
a
n
−
1
(
n
≥
2
)
,
a
0
=
a
1
=
1
. Find
a
n
a_n
a
n
.
14
1
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Find the Path
If
0
<
a
<
b
0<a<b
0
<
a
<
b
, given two fixed points
A
(
a
,
0
)
,
B
(
b
,
0
)
A(a,0),B(b,0)
A
(
a
,
0
)
,
B
(
b
,
0
)
. Draw lines
l
l
l
passes
A
A
A
,
m
m
m
passes
B
B
B
. They have four different intersections with parabola
y
2
=
x
y^2=x
y
2
=
x
. If the four points are concyclic, find the path of
P
(
P
=
l
∩
m
)
P(P=l\cap m)
P
(
P
=
l
∩
m
)
.
13
1
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Triangular Pyramid Problem
In triangular pyramid
S
−
A
B
C
S-ABC
S
−
A
BC
, any two of
S
A
,
S
B
,
S
C
SA,SB,SC
S
A
,
SB
,
SC
are perpendicular.
M
M
M
is the centre of gravity of
△
A
B
C
\triangle ABC
△
A
BC
.
D
D
D
is the midpoint of
A
B
AB
A
B
, line
D
P
/
/
S
C
DP//SC
D
P
//
SC
. Prove: (a)
D
P
DP
D
P
and
S
M
SM
SM
intersect. (b)
D
P
∩
S
M
=
D
′
DP\cap SM=D'
D
P
∩
SM
=
D
′
, then
D
′
D'
D
′
is the center of circumsphere of
S
−
A
B
C
S-ABC
S
−
A
BC
.
12
1
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900 Numbers
There are 900 3-digit-numbers
100
,
101
,
⋯
,
999
100,101,\cdots,999
100
,
101
,
⋯
,
999
. Print them on 900 cards. Some numbers are still numbers when we turn upside down, for example
198
198
198
is
861
861
861
when we turn upside down. However, some numbers are not, for example
531
531
531
. So, some numbers cam be used twice. If we want to express all 3-digit-numbers, we can print________cards fewer.
11
1
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Logarithm
Four real numbers
x
0
>
x
1
>
x
2
>
x
3
>
0
x_0>x_1>x_2>x_3>0
x
0
>
x
1
>
x
2
>
x
3
>
0
, if
log
x
0
x
1
1993
+
log
x
1
x
2
1993
+
log
x
2
x
3
1993
≥
k
⋅
log
x
0
x
3
1993
\log_{\frac{x_0}{x_1}}1993+\log_{\frac{x_1}{x_2}}1993+\log_{\frac{x_2}{x_3}}1993\geq k\cdot\log_{\frac{x_0}{x_3}}1993
lo
g
x
1
x
0
1993
+
lo
g
x
2
x
1
1993
+
lo
g
x
3
x
2
1993
≥
k
⋅
lo
g
x
3
x
0
1993
for all
x
0
,
x
1
,
x
2
,
x
3
x_0,x_1,x_2,x_3
x
0
,
x
1
,
x
2
,
x
3
, then the maximum value of
k
k
k
is________.
10
1
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The Last Two Digits of Number
The last two digits of number of
[
1
0
93
1
0
31
+
1
]
\left[\frac{10^{93}}{10^{31}+1}\right]
[
1
0
31
+
1
1
0
93
]
is________.
9
1
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Complex Number
If
z
∈
C
,
arg
(
z
2
−
4
)
=
5
6
π
,
arg
(
z
2
+
4
)
=
π
3
z\in\mathbb{C},\arg{(z^2-4)}=\frac{5}{6}\pi,\arg{(z^2+4)}=\frac{\pi}{3}
z
∈
C
,
ar
g
(
z
2
−
4
)
=
6
5
π
,
ar
g
(
z
2
+
4
)
=
3
π
, then the value of
z
z
z
is________.
8
1
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Max and Min
Real number
x
,
y
x,y
x
,
y
satisfy that
4
x
2
−
5
x
y
+
4
y
2
=
5
,
S
=
x
2
+
y
2
4x^2-5xy+4y^2=5,S=x^2+y^2
4
x
2
−
5
x
y
+
4
y
2
=
5
,
S
=
x
2
+
y
2
, then
1
S
max
+
1
S
min
=
\frac{1}{S_\text{max}}+\frac{1}{S_\text{min}}=
S
max
1
+
S
min
1
=
________.
7
1
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Quadratic Equation
Equation
(
1
−
i
)
x
2
+
(
λ
+
i
)
x
+
(
1
+
i
λ
)
=
0
(
λ
∈
R
)
(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})
(
1
−
i
)
x
2
+
(
λ
+
i
)
x
+
(
1
+
i
λ
)
=
0
(
λ
∈
R
)
has two imaginary roots, then the range value of
λ
\lambda
λ
is________.
6
1
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Choose the Figure in Complex Plane
m
,
n
m,n
m
,
n
are non-zero-real numbers,
z
∈
C
z\in\mathbb{C}
z
∈
C
. Then, the figure of equations
∣
z
+
n
i
∣
+
∣
z
−
m
i
∣
=
n
|z+n\text{i}|+|z-m\text{i}|=n
∣
z
+
n
i
∣
+
∣
z
−
m
i
∣
=
n
and
∣
z
+
n
i
∣
−
∣
z
−
m
i
∣
=
−
m
|z+n\text{i}|-|z-m\text{i}|=-m
∣
z
+
n
i
∣
−
∣
z
−
m
i
∣
=
−
m
in complex plane is (
F
1
,
F
2
F_1,F_2
F
1
,
F
2
are focal points) https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS84L2RkYWZjM2JmNTc0N2RmYjJlMGUwMGFmMWRkY2RkZTA4NTljZTUwLnBuZw==&rn=MTI0NTI0NTQucG5n
5
1
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Solve the Triangle
In
△
A
B
C
\triangle ABC
△
A
BC
,
c
−
a
c-a
c
−
a
is equal to height on side
A
C
AC
A
C
. Then, the value of
sin
C
−
A
2
+
cos
C
+
A
2
\sin\frac{C-A}{2}+\cos\frac{C+A}{2}
sin
2
C
−
A
+
cos
2
C
+
A
is
(A)
1
(B)
1
2
(C)
1
3
(D)
−
1
\text{(A)}1\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}-1
(A)
1
(B)
2
1
(C)
3
1
(D)
−
1
4
1
Hide problems
Analytic Geometry Problem
C
:
(
x
−
arcsin
a
)
(
x
−
arccos
a
)
+
(
y
−
arcsin
a
)
(
y
+
arccos
a
)
=
0
C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0
C
:
(
x
−
arcsin
a
)
(
x
−
arccos
a
)
+
(
y
−
arcsin
a
)
(
y
+
arccos
a
)
=
0
. The length of string of
C
C
C
cut by
l
:
x
=
π
4
l:x=\frac{\pi}{4}
l
:
x
=
4
π
is
d
d
d
. When
a
a
a
changes, the minumum value of
d
d
d
is
(A)
π
4
(B)
π
3
(C)
π
2
(D)
π
\text{(A)}\frac{\pi}{4}\qquad\text{(B)}\frac{\pi}{3}\qquad\text{(C)}\frac{\pi}{2}\qquad\text{(D)}\pi
(A)
4
π
(B)
3
π
(C)
2
π
(D)
π
3
2
Hide problems
The Number of Sets of Sets
Sets
A
,
B
A,B
A
,
B
satisfy that
A
∪
B
=
{
a
1
,
a
2
,
a
3
}
A\cup B=\{a_1,a_2,a_3\}
A
∪
B
=
{
a
1
,
a
2
,
a
3
}
. If
A
≠
B
A\neq B
A
=
B
, then
(
A
,
B
)
(A,B)
(
A
,
B
)
is different from
(
B
,
A
)
(B,A)
(
B
,
A
)
. The number of such sets
(
A
,
B
)
(A,B)
(
A
,
B
)
is
(A)
8
(B)
9
(C)
26
(D)
27
\text{(A)}8\qquad\text{(B)}9\qquad\text{(C)}26\qquad\text{(D)}27
(A)
8
(B)
9
(C)
26
(D)
27
Geometry
Horizontal line
m
m
m
passes the center of circle
⊙
O
\odot O
⊙
O
. Line
l
⊥
m
l\perp m
l
⊥
m
,
l
l
l
and
m
m
m
intersect at
M
M
M
, and
M
M
M
is on the right side of
O
O
O
. Three points
A
,
B
,
C
A,B,C
A
,
B
,
C
(
B
B
B
is in the middle) lie on line
l
l
l
, which are outside the circle, above line
m
m
m
.
A
P
,
B
Q
,
C
R
AP,BQ,CR
A
P
,
BQ
,
CR
are tangent to
⊙
O
\odot O
⊙
O
at
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
. Prove: (a) If
l
l
l
is tangent to
⊙
O
\odot O
⊙
O
, then
A
B
⋅
C
R
+
B
C
⋅
A
P
=
A
C
⋅
B
Q
AB\cdot CR+BC\cdot AP=AC\cdot BQ
A
B
⋅
CR
+
BC
⋅
A
P
=
A
C
⋅
BQ
. (b) If
l
l
l
and
⊙
O
\odot O
⊙
O
intersect, then
A
B
⋅
C
R
+
B
C
⋅
A
P
<
A
C
⋅
B
Q
AB\cdot CR+BC\cdot AP<AC\cdot BQ
A
B
⋅
CR
+
BC
⋅
A
P
<
A
C
⋅
BQ
. (c) If
l
l
l
and
⊙
O
\odot O
⊙
O
are apart, then
A
B
⋅
C
R
+
B
C
⋅
A
P
>
A
C
⋅
B
Q
AB\cdot CR+BC\cdot AP>AC\cdot BQ
A
B
⋅
CR
+
BC
⋅
A
P
>
A
C
⋅
BQ
.
2
2
Hide problems
A Function
f
(
x
)
=
a
sin
x
+
b
x
3
+
4
f(x)=a\sin x+b\sqrt[3]{x}+4
f
(
x
)
=
a
sin
x
+
b
3
x
+
4
. If
f
(
lg
log
3
10
)
=
5
f(\lg\log_{3}10)=5
f
(
l
g
lo
g
3
10
)
=
5
, then the value of
f
(
lg
lg
3
)
f(\lg\lg 3)
f
(
l
g
l
g
3
)
is
(A)
−
5
(B)
−
3
(C)
3
(D)
\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}
(A)
−
5
(B)
−
3
(C)
3
(D)
not sure
Problem of a Set
Set
∣
A
∣
=
n
|A|=n
∣
A
∣
=
n
.
A
1
,
A
2
,
⋯
,
A
m
A_1,A_2,\cdots,A_m
A
1
,
A
2
,
⋯
,
A
m
are subsets of
A
A
A
, and
A
i
⊈
A
j
A_i\not\subseteq A_j
A
i
⊆
A
j
for any
1
≤
i
<
j
≤
m
1\leq i<j\leq m
1
≤
i
<
j
≤
m
. Prove: (a)
∑
i
=
1
m
1
C
n
∣
A
i
∣
≤
1
\sum_{i=1}^{m}\frac{1}{\text{C}_n^{|A_i|}}\leq1
∑
i
=
1
m
C
n
∣
A
i
∣
1
≤
1
. (b)
∑
i
=
1
m
C
n
∣
A
i
∣
≥
m
2
\sum_{i=1}^{m}\text{C}_n^{|A_i|}\geq m^2
∑
i
=
1
m
C
n
∣
A
i
∣
≥
m
2
.
1
2
Hide problems
Problem of "0+0=0"
If
M
=
{
(
x
,
y
)
∣
∣
tan
π
x
∣
+
sin
2
π
x
=
0
}
,
N
=
{
(
x
,
y
)
∣
x
2
+
y
2
≤
2
}
M=\{(x,y)||\tan\pi x|+\sin^2\pi x=0\},N=\{(x,y)|x^2+y^2\leq2\}
M
=
{(
x
,
y
)
∣∣
tan
π
x
∣
+
sin
2
π
x
=
0
}
,
N
=
{(
x
,
y
)
∣
x
2
+
y
2
≤
2
}
, then
∣
M
∩
N
∣
|M\cap N|
∣
M
∩
N
∣
is equal to
(A)
4
(B)
5
(C)
8
(D)
9
\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}8\qquad\text{(D)}9
(A)
4
(B)
5
(C)
8
(D)
9
Simple Geometry
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, only
D
D
D
is an obtuse angle. Use some line segments to divide it into
n
n
n
obtuse triangles. But on its sides (except
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
), there is no vertex of triangles we divided into. Prove that if and only if
n
≥
4
n\geq4
n
≥
4
, we can divide the convex quadrilateral into such
n
n
n
triangles.