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Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1991 National High School Mathematics League
1991 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Inequality
If
0
<
a
<
1
,
x
2
+
y
=
0
0<a<1,x^2+y=0
0
<
a
<
1
,
x
2
+
y
=
0
, prove that
log
a
(
a
x
+
a
y
)
≤
log
a
2
+
1
8
\log_a(a^x+a^y)\leq\log_a2+\frac{1}{8}
lo
g
a
(
a
x
+
a
y
)
≤
lo
g
a
2
+
8
1
.
14
1
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Parabola Problem
O
O
O
is the vertex of a parabola,
F
F
F
is its focus.
P
Q
PQ
PQ
is a chord of the parabola. If
∣
O
F
∣
=
a
,
∣
P
Q
∣
=
b
|OF|=a,|PQ|=b
∣
OF
∣
=
a
,
∣
PQ
∣
=
b
, find the area of
△
O
P
Q
\triangle OPQ
△
OPQ
.
13
1
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Regular Triangular Pyramid Problem
In regular triangular pyramid
P
−
A
B
C
P-ABC
P
−
A
BC
,
P
O
PO
PO
is its height,
M
M
M
is the midpoint of
P
O
PO
PO
. Draw the plane that passes
A
M
AM
A
M
and parallel to
B
C
BC
BC
. Now the triangular pyramid is divided into two parts. Find the ratio of their volume.
12
1
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Problem of a Set
Set
M
=
{
1
,
2
,
⋯
,
1000
}
M=\{1,2,\cdots,1000\}
M
=
{
1
,
2
,
⋯
,
1000
}
, for any
X
⊆
M
(
X
≠
∅
)
X\subseteq M(X\neq\varnothing)
X
⊆
M
(
X
=
∅
)
, define
a
X
a_X
a
X
: sum of the minumum and maximum number in
X
X
X
. Then, the arithmetic mean of all
a
X
a_X
a
X
is________.
11
1
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Complex Numbers
For two complex numbers
z
1
,
z
2
z_1,z_2
z
1
,
z
2
satisfy that
∣
z
1
∣
=
∣
z
1
+
z
2
∣
=
3
,
∣
z
1
−
z
2
∣
=
3
3
|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3
∣
z
1
∣
=
∣
z
1
+
z
2
∣
=
3
,
∣
z
1
−
z
2
∣
=
3
3
, then
log
3
∣
(
z
1
z
2
‾
)
2000
+
(
z
1
‾
z
2
)
2000
∣
=
\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=
lo
g
3
∣
(
z
1
z
2
)
2000
+
(
z
1
z
2
)
2000
∣
=
________.
10
1
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Number Theory Problem
The remainder of
199
1
2000
1991^{2000}
199
1
2000
module
1
0
6
10^6
1
0
6
is________.
9
1
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Divide Odd Numbers Into Groups
Devide all odd numbers from small to large into groups: there are
2
n
−
1
2n-1
2
n
−
1
numbers in the
n
n
n
th group. For example: the first group is
{
1
}
\{1\}
{
1
}
, the second group is
{
3
,
5
,
7
}
\{3,5,7\}
{
3
,
5
,
7
}
, the third group is
{
9
,
11
,
13
,
15
,
17
}
\{9,11,13,15,17\}
{
9
,
11
,
13
,
15
,
17
}
. Then,
1991
1991
1991
is in group________.
8
1
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Triangle Problem
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
,
B
,
C
A,B,C
A
,
B
,
C
are arithmetic sequence, and
c
−
a
c-a
c
−
a
is equal to height on side
B
C
BC
BC
, then
sin
C
−
A
2
=
\sin\frac{C-A}{2}=
sin
2
C
−
A
=
________.
7
1
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Calculation
cos
2
1
0
∘
+
cos
2
50
−
sin
4
0
∘
⋅
sin
8
0
∘
\cos^210^{\circ}+\cos^250-\sin40^{\circ}\cdot\sin80^{\circ}
cos
2
1
0
∘
+
cos
2
50
−
sin
4
0
∘
⋅
sin
8
0
∘
=________.
6
1
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Choose the Figure
The figure of equation
∣
x
−
y
2
∣
=
1
−
∣
x
∣
|x-y^2|=1-|x|
∣
x
−
y
2
∣
=
1
−
∣
x
∣
is https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi80LzQ4YjgxN2YxMjc0YTBkNzZiZjJiMTRhMjBiNDExN2I5OGZhZGY3LnBuZw==&rn=MjAwMDAwMDAwMDAwMC5wbmc=
5
1
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Two Sets Again
S
=
{
(
x
,
y
)
∣
x
2
−
y
2
is odd
,
x
,
y
∈
R
}
,
T
=
{
(
x
,
y
)
∣
sin
(
2
π
x
2
)
−
sin
(
2
π
y
2
)
=
cos
(
2
π
x
2
)
−
cos
(
2
π
y
2
)
,
x
,
y
∈
R
}
S=\{(x,y)|x^2-y^2 \text{is odd},x,y\in\mathbb{R}\},T=\{(x,y)|\sin(2\pi x^2)-\sin(2\pi y^2)=\cos(2\pi x^2)-\cos(2\pi y^2),x,y\in\mathbb{R}\}
S
=
{(
x
,
y
)
∣
x
2
−
y
2
is odd
,
x
,
y
∈
R
}
,
T
=
{(
x
,
y
)
∣
sin
(
2
π
x
2
)
−
sin
(
2
π
y
2
)
=
cos
(
2
π
x
2
)
−
cos
(
2
π
y
2
)
,
x
,
y
∈
R
}
, then
(A)
S
⊂
T
(B)
T
⊂
S
(C)
S
=
T
(D)
S
∩
T
=
∅
\text{(A)}S\subset T\qquad\text{(B)}T\subset S\qquad\text{(C)}S=T\qquad\text{(D)}S\cap T=\varnothing
(A)
S
⊂
T
(B)
T
⊂
S
(C)
S
=
T
(D)
S
∩
T
=
∅
4
1
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Sum of Real Roots
Function
f
(
x
)
f(x)
f
(
x
)
satisfies that
f
(
3
+
x
)
=
f
(
3
−
x
)
f(3+x)=f(3-x)
f
(
3
+
x
)
=
f
(
3
−
x
)
. Also, equation
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
has six different real roots, then the sum of these roots is
(A)
18
(B)
12
(C)
9
(D)
0
\text{(A)}18\qquad\text{(B)}12\qquad\text{(C)}9\qquad\text{(D)}0
(A)
18
(B)
12
(C)
9
(D)
0
3
2
Hide problems
Number Theory
Let
a
a
a
be a positive integer,
a
<
100
a<100
a
<
100
, and
a
3
+
23
a^3+23
a
3
+
23
is a multiple of
24
24
24
. Then, the number of such
a
a
a
is
(A)
4
(B)
5
(C)
9
(D)
10
\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}9\qquad\text{(D)}10
(A)
4
(B)
5
(C)
9
(D)
10
Number Theory Problem
Let
a
n
a_n
a
n
be the number of such numbers
N
N
N
: sum of all digits of
N
N
N
is
n
n
n
, and each digit can only be
1
,
3
,
4
1,3,4
1
,
3
,
4
. Prove that
a
2
n
a_{2n}
a
2
n
is a perfect square for all
n
∈
Z
+
n\in\mathbb{Z}_+
n
∈
Z
+
.
2
2
Hide problems
Complex Number
a
,
b
,
c
a,b,c
a
,
b
,
c
are three non-zero-complex numbers, and
a
b
=
b
c
=
c
a
\frac{a}{b}=\frac{b}{c}=\frac{c}{a}
b
a
=
c
b
=
a
c
, then the value of
a
+
b
−
c
a
−
b
+
c
\frac{a+b-c}{a-b+c}
a
−
b
+
c
a
+
b
−
c
is (
ω
=
−
1
2
+
3
2
i
\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}
ω
=
−
2
1
+
2
3
i
)
(A)
1
(B)
±
ω
(C)
1
,
ω
,
ω
2
(D)
1
,
−
ω
,
−
ω
2
\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2
(A)
1
(B)
±
ω
(C)
1
,
ω
,
ω
2
(D)
1
,
−
ω
,
−
ω
2
Convex Quadrilateral
Area of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is
1
1
1
. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than
1
4
\frac{1}{4}
4
1
.
1
2
Hide problems
Count the Triangles
The number of regular triangles that three apexes are among eight vertex of a cube is
(A)
4
(B)
8
(C)
12
(D)
24
\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24
(A)
4
(B)
8
(C)
12
(D)
24
Count the Number of Arithmetic Sequences
Set
S
=
{
1
,
2
,
⋯
,
n
}
S=\{1,2,\cdots,n\}
S
=
{
1
,
2
,
⋯
,
n
}
.
A
A
A
is an increasing arithmetic sequence (at least two numbers), and all numbers are in
S
S
S
. Also, we can't add any number in
S
S
S
to
A
A
A
without changing its tolerance. Find the number of such sequence
A
A
A
.