MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1982 National High School Mathematics League
1982 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(12)
12
1
Hide problems
Geometry Or Number Theory ?
Given a circle
C
:
x
2
+
y
2
=
r
2
C:x^2+y^2=r^2
C
:
x
2
+
y
2
=
r
2
(
r
r
r
is an odd number).
P
(
u
,
v
)
∈
C
P(u,v)\in C
P
(
u
,
v
)
∈
C
, satisfying:
u
=
p
m
,
v
=
q
n
u=p^m, v=q^n
u
=
p
m
,
v
=
q
n
(
p
,
q
p,q
p
,
q
are prime numbers,
m
,
n
m,n
m
,
n
are integers,
u
>
v
u>v
u
>
v
). Define
A
,
B
,
C
,
D
,
M
,
N
:
A
(
r
,
0
)
,
B
(
−
r
,
0
)
,
C
(
0
,
−
r
)
,
D
(
0
,
r
)
,
M
(
u
,
0
)
,
N
(
0
,
v
)
A,B,C,D,M,N:A(r,0),B(-r,0),C(0,-r),D(0,r),M(u,0),N(0,v)
A
,
B
,
C
,
D
,
M
,
N
:
A
(
r
,
0
)
,
B
(
−
r
,
0
)
,
C
(
0
,
−
r
)
,
D
(
0
,
r
)
,
M
(
u
,
0
)
,
N
(
0
,
v
)
. Prove that
∣
A
M
∣
=
1
,
∣
B
M
∣
=
9
,
∣
C
N
∣
=
8
,
∣
D
N
∣
=
2
|AM|=1,|BM|=9,|CN|=8,|DN|=2
∣
A
M
∣
=
1
,
∣
BM
∣
=
9
,
∣
CN
∣
=
8
,
∣
D
N
∣
=
2
.
11
1
Hide problems
Geometry Calculation
Length of edges of regular triangle
A
B
C
ABC
A
BC
are
4
4
4
,
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
, satisfying:
∣
A
E
∣
=
∣
B
F
∣
=
∣
C
D
∣
=
1
|AE|=|BF|=|CD|=1
∣
A
E
∣
=
∣
BF
∣
=
∣
C
D
∣
=
1
.
B
E
∩
C
F
=
R
,
C
F
∩
A
D
=
Q
,
A
D
∩
B
E
=
S
BE\cap CF=R, CF\cap AD=Q, AD\cap BE=S
BE
∩
CF
=
R
,
CF
∩
A
D
=
Q
,
A
D
∩
BE
=
S
.
P
P
P
is a point inside
△
R
Q
S
\triangle RQS
△
RQS
or on its sides. Note that
x
=
d
(
P
,
B
C
)
,
y
=
d
(
P
,
C
A
)
,
z
=
d
(
P
,
A
B
)
x=d(P,BC),y=d(P,CA),z=d(P,AB)
x
=
d
(
P
,
BC
)
,
y
=
d
(
P
,
C
A
)
,
z
=
d
(
P
,
A
B
)
. (a)
x
y
z
xyz
x
yz
get its minumum value when
P
=
R
P=R
P
=
R
(or
Q
,
S
Q,S
Q
,
S
). (b) Calculate the minumum value of
x
y
z
xyz
x
yz
.
10
1
Hide problems
Simple Geometry
Semi-circle
A
B
AB
A
B
with diameter
A
B
AB
A
B
, and
A
B
=
2
r
AB=2r
A
B
=
2
r
. Given line
l
l
l
, satisfying that
l
⊥
B
A
,
l
∩
B
A
=
T
,
∣
A
T
∣
=
2
a
(
2
a
<
r
)
l \perp BA, l \cap BA=T , |AT|=2a(2a<r)
l
⊥
B
A
,
l
∩
B
A
=
T
,
∣
A
T
∣
=
2
a
(
2
a
<
r
)
.
M
,
N
M,N
M
,
N
are two points on the semi-circle, such that
d
(
M
,
l
)
=
∣
A
M
∣
,
d
(
N
,
l
)
=
∣
A
N
∣
(
M
≠
N
)
.
d(M,l)=|AM|,d(N,l)=|AN|(M\neq N).
d
(
M
,
l
)
=
∣
A
M
∣
,
d
(
N
,
l
)
=
∣
A
N
∣
(
M
=
N
)
.
Prove:
∣
A
M
∣
+
∣
A
N
∣
=
∣
A
B
∣
|AM|+|AN|=|AB|
∣
A
M
∣
+
∣
A
N
∣
=
∣
A
B
∣
.
9
1
Hide problems
3D Geometry
In tetrahedron
S
A
B
C
SABC
S
A
BC
,
∠
A
S
B
=
π
2
,
∠
A
S
C
=
α
(
0
<
α
<
π
2
)
,
∠
B
S
C
=
β
(
0
<
β
<
π
2
)
\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})
∠
A
SB
=
2
π
,
∠
A
SC
=
α
(
0
<
α
<
2
π
)
,
∠
BSC
=
β
(
0
<
β
<
2
π
)
. Let
θ
=
A
−
S
C
−
B
\theta=A-SC-B
θ
=
A
−
SC
−
B
, prove that
θ
=
−
arccos
(
cot
α
⋅
cot
β
)
\theta=-\arccos(\cot\alpha\cdot\cot\beta)
θ
=
−
arccos
(
cot
α
⋅
cot
β
)
.
8
1
Hide problems
Simple Inequality
a
,
b
a,b
a
,
b
are two different positive real numbers, then which one is the largest?
A
=
(
a
+
1
a
)
(
b
+
1
b
)
,
B
=
(
a
b
+
1
a
b
)
2
,
C
=
(
a
+
b
2
+
2
a
+
b
)
2
.
A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.
A
=
(
a
+
a
1
)
(
b
+
b
1
)
,
B
=
(
ab
+
ab
1
)
2
,
C
=
(
2
a
+
b
+
a
+
b
2
)
2
.
(A)
A
(B)
B
(C)
C
(D)
\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}
(A)
A
(B)
B
(C)
C
(D)
Not sure.
7
1
Hide problems
Two magical sets
Let
M
=
{
(
x
,
y
)
∣
∣
x
y
∣
=
1
,
x
>
0
}
,
N
=
{
(
x
,
y
)
∣
arctan
x
+
arctan
y
=
π
}
M=\{(x,y)||xy|=1,x>0\},N=\{(x,y)|\arctan x+\arctan y=\pi\}
M
=
{(
x
,
y
)
∣∣
x
y
∣
=
1
,
x
>
0
}
,
N
=
{(
x
,
y
)
∣
arctan
x
+
arctan
y
=
π
}
. Which one is right?
(A)
M
∪
N
=
{
(
x
,
y
)
∣
∣
x
y
∣
=
1
}
(B)
M
∪
N
=
M
\text{(A)}M\cup N=\{(x,y)||xy|=1\}\qquad\text{(B)}M\cup N=M
(A)
M
∪
N
=
{(
x
,
y
)
∣∣
x
y
∣
=
1
}
(B)
M
∪
N
=
M
(C)
M
∪
N
=
N
(D)
M
∪
N
=
{
(
x
,
y
)
∣
∣
x
y
∣
=
1
,
x
,
y
cannot be negative the same time
}
\text{(C)}M\cup N=N\qquad\text{(D)}M\cup N=\{(x,y)||xy|=1,x,y\text{ cannot be negative the same time}\}
(C)
M
∪
N
=
N
(D)
M
∪
N
=
{(
x
,
y
)
∣∣
x
y
∣
=
1
,
x
,
y
cannot be negative the same time
}
6
1
Hide problems
Vieta theorem
x
1
,
x
2
x_1,x_2
x
1
,
x
2
are two real roots of the equation
x
2
−
(
k
−
2
)
x
+
(
k
2
+
3
k
+
5
)
=
0
x^2-(k-2)x+(k^2+3k+5)=0
x
2
−
(
k
−
2
)
x
+
(
k
2
+
3
k
+
5
)
=
0
.What's the maximum value of
x
1
2
+
x
2
2
x_1^2+x_2^2
x
1
2
+
x
2
2
?
(A)
19
(B)
18
(C)
5
5
9
(D)
\text{(A)}19\qquad\text{(B)}18\qquad\text{(C)}5\frac{5}{9}\qquad\text{(D)}
(A)
19
(B)
18
(C)
5
9
5
(D)
Not exist
5
1
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Trigonometric functions
For any
φ
∈
(
0
,
π
2
)
\varphi\in(0,\frac{\pi}{2})
φ
∈
(
0
,
2
π
)
, we have
(A)
sin
sin
φ
<
cos
φ
<
cos
cos
φ
(B)
sin
sin
φ
>
cos
φ
>
cos
cos
φ
\text{(A)}\sin\sin\varphi<\cos\varphi<\cos\cos\varphi\qquad\text{(B)}\sin\sin\varphi>\cos\varphi>\cos\cos\varphi
(A)
sin
sin
φ
<
cos
φ
<
cos
cos
φ
(B)
sin
sin
φ
>
cos
φ
>
cos
cos
φ
(C)
sin
cos
φ
>
cos
φ
>
cos
sin
φ
(D)
sin
cos
φ
<
cos
φ
<
cos
sin
φ
\text{(C)}\sin\cos\varphi>\cos\varphi>\cos\sin\varphi\qquad\text{(D)}\sin\cos\varphi<\cos\varphi<\cos\sin\varphi
(C)
sin
cos
φ
>
cos
φ
>
cos
sin
φ
(D)
sin
cos
φ
<
cos
φ
<
cos
sin
φ
4
1
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Calculate the Area
What's the area defined by equation
∣
x
−
1
∣
+
∣
y
−
1
∣
=
1
|x-1|+|y-1|=1
∣
x
−
1∣
+
∣
y
−
1∣
=
1
?
(A)
1
(B)
2
(C)
π
(D)
4
\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}\pi\qquad\text{(D)}4
(A)
1
(B)
2
(C)
π
(D)
4
3
1
Hide problems
What a Mess!
If
log
2
(
log
1
2
(
log
2
x
)
)
=
log
3
(
log
1
3
(
log
3
y
)
)
=
log
5
(
log
1
5
(
log
5
z
)
)
=
0
\log_2(\log_{\frac{1}{2}}(\log_2x))=\log_3(\log_{\frac{1}{3}}(\log_3y))=\log_5(\log_{\frac{1}{5}}(\log_5z))=0
lo
g
2
(
lo
g
2
1
(
lo
g
2
x
))
=
lo
g
3
(
lo
g
3
1
(
lo
g
3
y
))
=
lo
g
5
(
lo
g
5
1
(
lo
g
5
z
))
=
0
, then
(A)
z
<
x
<
y
(B)
x
<
y
<
z
(C)
y
<
z
<
x
(D)
z
<
y
<
x
\text{(A)}z<x<y\qquad\text{(B)}x<y<z\qquad\text{(C)}y<z<x\qquad\text{(D)}z<y<x
(A)
z
<
x
<
y
(B)
x
<
y
<
z
(C)
y
<
z
<
x
(D)
z
<
y
<
x
2
1
Hide problems
Polar Coordinates
In polar coordinates, the equation
ρ
=
1
1
−
cos
θ
+
sin
θ
\rho=\frac{1}{1-\cos\theta+\sin\theta}
ρ
=
1
−
c
o
s
θ
+
s
i
n
θ
1
stands for a
(A)
\text{(A)}
(A)
circle
(B)
\text{(B)}
(B)
ellipse
(C)
\text{(C)}
(C)
hyperbola
(D)
\text{(D)}
(D)
parabola
1
1
Hide problems
Is this a geometry problem or not?
For a convex polygon with
n
n
n
edges
F
F
F
, if all its diagonals have the equal length, then
(A)
F
∈
{
quadrilaterals
}
\text{(A)}F\in \{\text{quadrilaterals}\}
(A)
F
∈
{
quadrilaterals
}
(B)
F
∈
{
pentagons
}
\text{(B)}F\in \{\text{pentagons}\}
(B)
F
∈
{
pentagons
}
(C)
F
∈
{
pentagons
}
∪
{
quadrilaterals
}
\text{(C)}F\in \{\text{pentagons}\} \cup\{\text{quadrilaterals}\}
(C)
F
∈
{
pentagons
}
∪
{
quadrilaterals
}
(D)
F
∈
{
convex polygons that have all edges’ length equal
}
∪
{
convex polygons that have all inner angles equal
}
\text{(D)}F\in \{\text{convex polygons that have all edges' length equal}\} \cup\{\text{convex polygons that have all inner angles equal}\}
(D)
F
∈
{
convex polygons that have all edges’ length equal
}
∪
{
convex polygons that have all inner angles equal
}