MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
1987 National High School Mathematics League
1987 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(9)
9
1
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An Interesting Activity
Five sets of brothers and sisters attend an activity of
k
k
k
groups, stipulate that: (1)Anyone cannot be in the same group with his/her sister/brother. (2)Anyone has been in the same group with any other people who is not his/her sister/brother. (3)Only one person has attended moe than one group. Then, the minimun value of
k
k
k
is________.
8
1
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How Many Tetrahedrons?
We have two triangles that lengths of its sides are
3
,
4
,
5
3,4,5
3
,
4
,
5
, one triangle that lengths of its sides are
4
,
5
,
41
4,5,\sqrt{41}
4
,
5
,
41
, one triangle that lengths of its sides are
5
6
2
,
4
,
5
\frac{5}{6}\sqrt2,4,5
6
5
2
,
4
,
5
. The number of tetrahedrons with such four surfaces is________.
7
1
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Last digit number
k
(
k
>
1
)
k(k>1)
k
(
k
>
1
)
is an integer, and
a
a
a
is a solution to the equation
x
2
−
k
x
+
1
=
0
x^2-kx+1=0
x
2
−
k
x
+
1
=
0
. For any integer
n
(
n
>
10
)
n(n>10)
n
(
n
>
10
)
, the last digit number of
a
2
n
+
a
−
2
n
a^{2^n}+a^{-2^n}
a
2
n
+
a
−
2
n
is always
7
7
7
, then the last digit number of
k
k
k
is________.
6
1
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Eight Vertices of a Octagon
Set
A
=
{
(
x
,
y
)
∣
∣
x
∣
+
∣
y
∣
=
a
,
a
>
0
}
,
B
=
{
(
x
,
y
)
∣
∣
x
y
∣
+
1
=
∣
x
∣
+
∣
y
∣
}
A=\{(x,y)||x|+|y|=a,a>0\},B=\{(x,y)||xy|+1=|x|+|y|\}
A
=
{(
x
,
y
)
∣∣
x
∣
+
∣
y
∣
=
a
,
a
>
0
}
,
B
=
{(
x
,
y
)
∣∣
x
y
∣
+
1
=
∣
x
∣
+
∣
y
∣
}
. If
A
∩
B
A\cap B
A
∩
B
is a set of eight vertices of a regular octagon, then
a
=
a=
a
=
________.
5
1
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Two Equal Set
Two sets
M
=
{
x
,
x
y
,
lg
(
x
y
)
}
,
N
=
{
0
,
∣
x
∣
,
y
}
M=\{x,xy,\lg(xy)\},N=\{0,|x|,y\}
M
=
{
x
,
x
y
,
l
g
(
x
y
)}
,
N
=
{
0
,
∣
x
∣
,
y
}
, if
M
=
N
M=N
M
=
N
, then
(
x
+
1
y
)
+
(
x
2
+
1
y
2
)
+
⋯
+
(
x
2001
+
1
y
2001
)
=
(x+\frac{1}{y})+(x^2+\frac{1}{y^2})+\cdots+(x^{2001}+\frac{1}{y^{2001}})=
(
x
+
y
1
)
+
(
x
2
+
y
2
1
)
+
⋯
+
(
x
2001
+
y
2001
1
)
=
________.
4
1
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Rotate Problem
B
B
B
is the center of unit circle.
A
,
C
A,C
A
,
C
are points on the circle (the order of
A
,
B
,
C
A,B,C
A
,
B
,
C
is clockwise), and
∠
A
B
C
=
2
α
(
0
<
α
<
π
3
)
\angle ABC=2\alpha(0<\alpha<\frac{\pi}{3})
∠
A
BC
=
2
α
(
0
<
α
<
3
π
)
. Then we will rotate
△
A
B
C
\triangle ABC
△
A
BC
anticlockwise. In the first rotation,
A
A
A
is the center of rotation, the result is that
B
B
B
is on the circle. In the second rotation,
B
B
B
is the center of rotation, the result is that
C
C
C
is on the circle. In the third rotation,
C
C
C
is the center of rotation, the result is that
A
A
A
is on the circle. ... After we rotate for
100
100
100
times, the distance
A
A
A
travelled is
(A)
22
π
(
1
+
sin
α
)
−
66
α
(B)
67
3
π
(C)
22
π
+
68
3
π
sin
α
−
66
α
(D)
33
π
−
66
α
\text{(A)}22\pi(1+\sin\alpha)-66\alpha\qquad\text{(B)}\frac{67}{3}\pi\qquad\text{(C)}22\pi+\frac{68}{3}\pi\sin\alpha-66\alpha\qquad\text{(D)}33\pi-66\alpha
(A)
22
π
(
1
+
sin
α
)
−
66
α
(B)
3
67
π
(C)
22
π
+
3
68
π
sin
α
−
66
α
(D)
33
π
−
66
α
3
2
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Rational Point Problem
In rectangular coordinate system, define that if and only if both
x
x
x
-axis and
y
y
y
-axis of a point are rational numbers, we call it rational point. If
a
a
a
is an irrational number, then in all lines that passes
(
a
,
0
)
(a,0)
(
a
,
0
)
,
(A)
\text{(A)}
(A)
There are infinitely many lines, on which there are at least two rational points.
(B)
\text{(B)}
(B)
There are exactly
n
(
n
≥
2
)
n(n\geq2)
n
(
n
≥
2
)
lines, on which there are at least two rational points.
(C)
\text{(C)}
(C)
There are exactly 1 line, on which there are at least two rational points.
(D)
\text{(D)}
(D)
Every line passes at least one rational point.
A Ping-pong Game
n
(
n
>
3
)
n(n>3)
n
(
n
>
3
)
ping-pong players have played a few ping-pong games. The set of players that player A has played with is
A
A
A
, The set of players that player B has played with is
B
B
B
. for any two players,
A
≠
B
A\neq B
A
=
B
. Prove that we can delete a player, so that this character remains.
2
2
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Rhombus Problem
For a rhombus with side length of 5, length of one of its diagonal is not larger than
6
6
6
, length of the other diagonal is not smaller than
6
6
6
, then the maximum value of the sum of the two diagonals is
(A)
10
2
(B)
14
(C)
5
6
(D)
12
\text{(A)}10\sqrt{2}\qquad\text{(B)}14\qquad\text{(C)}5\sqrt{6}\qquad\text{(D)}12
(A)
10
2
(B)
14
(C)
5
6
(D)
12
Something Strange...
In rectangular coordinate system, define that if and only if both
x
x
x
-axis and
y
y
y
-axis of a point are integers, we call it integral point. Prove that there exists a series of concentric circles, satisfying: (1)Exery itengral point is on the concentric circles. (2)On each circle, there is exactly one itengral point.
1
2
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Perfect Cube
For any given positive integer
n
n
n
,
n
6
+
3
a
n^6+3a
n
6
+
3
a
is a perfect cube, where
a
a
a
is a positive integer. Then
(A)
\text{(A)}
(A)
There is no such
a
a
a
.
(B)
\text{(B)}
(B)
There are infinitely many such
a
a
a
.
(C)
\text{(C)}
(C)
There is finitely many such
a
a
a
.
(D)
\text{(D)}
(D)
None of
(A)(B)(C)
\text{(A)(B)(C)}
(A)(B)(C)
is correct.
Geometry Problem
△
A
B
C
\triangle ABC
△
A
BC
and
△
A
D
E
\triangle ADE
△
A
D
E
(
∠
A
B
C
=
∠
A
D
E
=
π
2
)
(\angle ABC=\angle ADE=\frac{\pi}{2})
(
∠
A
BC
=
∠
A
D
E
=
2
π
)
are two isosceles right triangle that are not congruent. Fix
△
A
B
C
\triangle ABC
△
A
BC
, but rotate
△
A
D
E
\triangle ADE
△
A
D
E
on the plane. Prove that there exists point
M
∈
B
C
M\in BC
M
∈
BC
, satisfying that
△
B
M
D
\triangle BMD
△
BM
D
is an isosceles right triangle.