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Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1981 National High School Mathematics League
1981 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(11)
11
1
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Let's Play Billiards!
A billiards table is in the figure of regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
.
P
P
P
is the midpoint of
A
B
AB
A
B
. We shut the ball at
P
P
P
, then it touches
Q
Q
Q
on side
B
C
BC
BC
, then it touches side
C
D
,
D
E
,
E
F
,
F
A
CD,DE,EF,FA
C
D
,
D
E
,
EF
,
F
A
. Finally, the ball touches side
A
B
AB
A
B
again. Let
θ
=
∠
B
P
Q
\theta=\angle BPQ
θ
=
∠
BPQ
, find the value range of
θ
\theta
θ
.
10
1
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How can we make products?
To make three kinds of products
A
,
B
,
C
A,B,C
A
,
B
,
C
, we have three parts
a
,
b
,
c
a,b,c
a
,
b
,
c
. A product
A
A
A
is made of two
a
a
a
and two
b
b
b
; a product
B
B
B
is made of one
b
b
b
and one
c
c
c
; a product
C
C
C
is made of two
a
a
a
and one
c
c
c
. We have a few parts. If we make
p
p
p
product
A
A
A
,
q
q
q
product
B
B
B
,
r
r
r
product
C
C
C
, then
2
2
2
part
a
a
a
and
1
1
1
part
b
b
b
are remained. Prove: no matter how we make products, we cannot use up all the parts.
9
1
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Geometry Inequality
O
O
O
is a circle with a radius of
1
1
1
, with strings
C
D
CD
C
D
and
E
F
EF
EF
.
C
D
/
/
E
F
CD//EF
C
D
//
EF
, and diameter
A
B
AB
A
B
intersects
C
D
,
E
F
CD,EF
C
D
,
EF
at
P
,
Q
P,Q
P
,
Q
. If
∠
B
P
D
=
π
4
\angle BPD=\frac{\pi}{4}
∠
BP
D
=
4
π
, prove that
P
C
⋅
Q
E
+
P
D
⋅
Q
F
<
2.
PC\cdot QE+PD \cdot QF<2.
PC
⋅
QE
+
P
D
⋅
QF
<
2.
8
1
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Logarithm Table
In the logarithm table below, there are two mistakes. Correct them.\begin{tabular}{|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
lg
0.021
\lg0.021
l
g
0.021
&
2
a
+
b
+
c
−
3
2a+b+c-3
2
a
+
b
+
c
−
3
\\ \hline
lg
0.27
\lg0.27
l
g
0.27
&
6
a
−
3
b
−
2
6a-3b-2
6
a
−
3
b
−
2
\\ \hline
lg
1.5
\lg1.5
l
g
1.5
&
3
a
−
b
+
c
3a-b+c
3
a
−
b
+
c
\\ \hline
lg
2.8
\lg2.8
l
g
2.8
&
1
−
2
a
+
2
b
−
c
1-2a+2b-c
1
−
2
a
+
2
b
−
c
\\ \hline
lg
3
\lg3
l
g
3
&
2
a
−
b
2a-b
2
a
−
b
\\ \hline
lg
5
\lg5
l
g
5
&
a
+
c
a+c
a
+
c
\\ \hline
lg
6
\lg6
l
g
6
&
1
+
a
−
b
−
c
1+a-b-c
1
+
a
−
b
−
c
\\ \hline
lg
7
\lg7
l
g
7
&
2
(
a
+
c
)
2(a+c)
2
(
a
+
c
)
\\ \hline
lg
8
\lg8
l
g
8
&
3
−
3
a
−
3
c
3-3a-3c
3
−
3
a
−
3
c
\\ \hline
lg
9
\lg9
l
g
9
&
4
a
−
2
b
4a-2b
4
a
−
2
b
\\ \hline
lg
14
\lg14
l
g
14
&
1
−
a
+
2
b
1-a+2b
1
−
a
+
2
b
\\ \hline \end{tabular}
7
1
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An Equation
The equation
x
∣
x
∣
+
p
x
+
q
=
0
x|x|+px+q=0
x
∣
x
∣
+
p
x
+
q
=
0
is given. Which of the following is not true?
(A)
\text{(A)}
(A)
It has at most three real roots.
(B)
\text{(B)}
(B)
It has at least one real root.
(C)
\text{(C)}
(C)
Only if
p
2
−
4
q
≥
0
p^2-4q\geq0
p
2
−
4
q
≥
0
,it has real roots.
(D)
\text{(D)}
(D)
If
p
<
0
p<0
p
<
0
and
q
>
0
q>0
q
>
0
, it has three real roots.
6
1
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Cartesian coordinates
In Cartesian coordinates, two areas
M
,
N
M,N
M
,
N
are defined below:
M
:
y
≥
0
,
y
≤
x
,
y
≤
2
−
x
M:y\geq0,y\leq x,y\leq 2-x
M
:
y
≥
0
,
y
≤
x
,
y
≤
2
−
x
;
N
:
t
≤
x
≤
t
+
1
N:t\leq x\leq t+1
N
:
t
≤
x
≤
t
+
1
.
t
t
t
is a real number that
t
∈
[
0
,
1
]
t\in[0,1]
t
∈
[
0
,
1
]
. Then the area of
M
∩
N
M\cap N
M
∩
N
is
(A)
−
t
2
+
t
+
1
2
(B)
−
2
t
2
+
2
t
(C)
1
−
2
t
2
(D)
1
2
(
t
−
2
)
2
\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2
(A)
−
t
2
+
t
+
2
1
(B)
−
2
t
2
+
2
t
(C)
1
−
2
t
2
(D)
2
1
(
t
−
2
)
2
5
1
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You Should Learn to Count!
Given a cube
A
B
C
D
−
A
′
B
′
C
′
D
′
ABCD-A'B'C'D'
A
BC
D
−
A
′
B
′
C
′
D
′
, in the
12
12
12
lines:
A
B
′
,
B
A
′
,
C
D
′
,
D
C
′
,
A
D
′
,
D
A
′
,
B
C
′
,
C
B
′
,
A
C
,
B
D
,
A
′
C
′
,
B
′
D
′
AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'
A
B
′
,
B
A
′
,
C
D
′
,
D
C
′
,
A
D
′
,
D
A
′
,
B
C
′
,
C
B
′
,
A
C
,
B
D
,
A
′
C
′
,
B
′
D
′
, how many sets of lines are skew lines?
(A)
30
(B)
60
(C)
24
(D)
48
\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48
(A)
30
(B)
60
(C)
24
(D)
48
4
1
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Largest Area?
In the four figures, which one has the largest area?
(A)
△
A
B
C
:
∠
A
=
π
3
,
∠
B
=
π
4
,
∣
A
C
∣
=
2
\text{(A)}\triangle ABC: \angle A=\frac{\pi}{3},\angle B=\frac{\pi}{4},|AC|=\sqrt2
(A)
△
A
BC
:
∠
A
=
3
π
,
∠
B
=
4
π
,
∣
A
C
∣
=
2
(B)
\text{(B)}
(B)
trapezium: two diagonals are
2
\sqrt2
2
and
3
\sqrt3
3
, intersection angle is
5
π
12
\frac{5\pi}{12}
12
5
π
.
(C)
\text{(C)}
(C)
Circle: with a radius of
1
1
1
.
(D)
\text{(D)}
(D)
Square: the length of a diagonal is
2.5
2.5
2.5
.
3
1
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Trigonometric functions
Let
α
\alpha
α
be a real number and
α
≠
k
π
2
,
k
∈
Z
\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}
α
=
2
kπ
,
k
∈
Z
,
T
=
sin
α
+
tan
α
cos
α
+
cot
α
T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}
T
=
cos
α
+
cot
α
sin
α
+
tan
α
.
(A)
\text{(A)}
(A)
T
T
T
is negative.
(B)
\text{(B)}
(B)
T
T
T
is nonnegative.
(C)
\text{(C)}
(C)
T
T
T
is positive.
(D)
\text{(D)}
(D)
T
T
T
can be either positive or negative.
2
1
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The Second one, still easy!
Given two conditions: A:
1
+
sin
θ
=
a
\sqrt{1+\sin\theta}=a
1
+
sin
θ
=
a
B:
sin
θ
2
+
cos
θ
2
=
a
\sin\frac{\theta}{2}+\cos\frac{\theta}{2}=a
sin
2
θ
+
cos
2
θ
=
a
Then, which one of the followings are true?
(
A
)
(\text{A})
(
A
)
A is sufficient and necessary condition of B.
(
B
)
(\text{B})
(
B
)
A is necessary but insufficient condition of B.
(
C
)
(\text{C})
(
C
)
A is sufficient but unnecessary condition of B.
(
D
)
(\text{D})
(
D
)
A is insufficient and unnecessary condition of B.
1
1
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To Start With - Very Easy!
Given two conditions: A: Two triangles have the same area and two corresponding edge equal. B: Two triangles are congruent. Then, which one of the followings are true?
(
A
)
(\text{A})
(
A
)
A is sufficient and necessary condition of B.
(
B
)
(\text{B})
(
B
)
A is necessary but insufficient condition of B.
(
C
)
(\text{C})
(
C
)
A is sufficient but unnecessary condition of B.
(
D
)
(\text{D})
(
D
)
A is insufficient and unnecessary condition of B.