MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
2005 National High School Mathematics League
2005 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
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Analytic Geometry
A
(
1
,
1
)
A(1,1)
A
(
1
,
1
)
is a point on parabola
y
=
x
2
y=x^2
y
=
x
2
. Draw the tangent line of the parabola that passes
A
A
A
, the line intersects
x
x
x
-axis at
D
D
D
, intersects
y
y
y
-axis at
B
B
B
.
C
C
C
is a point on the parabola, and
E
E
E
is a point on segment
A
C
AC
A
C
, such that
A
E
E
C
=
λ
1
\frac{AE}{EC}=\lambda_1
EC
A
E
=
λ
1
,
F
F
F
is a point on segment
B
C
BC
BC
, such that
B
F
F
C
=
λ
2
\frac{BF}{FC}=\lambda_2
FC
BF
=
λ
2
. If
λ
1
+
λ
2
=
1
\lambda_1+\lambda_2=1
λ
1
+
λ
2
=
1
,
C
D
CD
C
D
and
E
F
EF
EF
intersect at
P
P
P
. When
C
C
C
moves, find the path equation of
P
P
P
.
14
1
Hide problems
Counting
Nine balls numbered
1
,
2
,
⋯
,
9
1,2,\cdots,9
1
,
2
,
⋯
,
9
are put on nine poines that divide the circle into nine equal parts. The sum of absolute values of the difference between the number of two adjacent balls is
S
S
S
. Find the probablity of
S
S
S
takes its minumum value. Note: If one way of putting balls can be the same as another one by rotating or specular-reflecting, then they are considered the same way.
13
1
Hide problems
Sequence
Define sequence
(
a
n
)
(a_n)
(
a
n
)
:
a
0
=
1
,
a
n
+
1
=
7
a
n
+
45
a
n
2
−
36
2
,
n
∈
N
a_0=1,a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2},n\in\mathbb{N}
a
0
=
1
,
a
n
+
1
=
2
7
a
n
+
45
a
n
2
−
36
,
n
∈
N
. (a) Prove that for all
n
∈
N
n\in\mathbb{N}
n
∈
N
,
a
n
a_n
a
n
is a positive integer. (b) Prove that for all
n
∈
N
n\in\mathbb{N}
n
∈
N
,
a
n
a
n
+
1
−
1
a_na_{n+1}-1
a
n
a
n
+
1
−
1
is a perfect square.
12
1
Hide problems
Order the Numbers
If the sum of all digits of a number is
7
7
7
, then we call it lucky number. Put all lucky numbers in order (from small to large):
a
1
,
a
2
,
⋯
,
a
n
,
⋯
a_1,a_2,\cdots,a_n,\cdots
a
1
,
a
2
,
⋯
,
a
n
,
⋯
. If
a
n
=
2005
a_n=2005
a
n
=
2005
, then
a
5
n
=
a_{5n}=
a
5
n
=
________.
11
1
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The Area of the Square
One side of a square in on line
y
=
2
x
−
17
y=2x-17
y
=
2
x
−
17
, and two other points are on parabola
y
=
x
2
y=x^2
y
=
x
2
, then the minumum value of the area of the square is________.
10
1
Hide problems
3D Geometry
In tetrahedron
A
B
C
D
ABCD
A
BC
D
, the volume of tetrahedron
A
B
C
D
ABCD
A
BC
D
is
1
6
\frac{1}{6}
6
1
, and
∠
A
C
B
=
4
5
∘
,
A
D
+
B
C
+
A
C
2
=
3
\angle ACB=45^{\circ},AD+BC+\frac{AC}{\sqrt2}=3
∠
A
CB
=
4
5
∘
,
A
D
+
BC
+
2
A
C
=
3
, then
C
D
=
CD=
C
D
=
________.
9
1
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Trigonometry
If
0
<
α
<
β
<
γ
<
2
π
0<\alpha<\beta<\gamma<2\pi
0
<
α
<
β
<
γ
<
2
π
, for all
x
∈
R
x\in\mathbb{R}
x
∈
R
,
cos
(
x
+
α
)
+
cos
(
x
+
β
)
+
cos
(
x
+
γ
)
=
0
\cos(x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0
cos
(
x
+
α
)
+
cos
(
x
+
β
)
+
cos
(
x
+
γ
)
=
0
, then
γ
−
α
=
\gamma-\alpha=
γ
−
α
=
________.
8
1
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A Function
f
(
x
)
f(x)
f
(
x
)
is a decreasing function defined on
(
0
,
+
∞
)
(0,+\infty)
(
0
,
+
∞
)
, if
f
(
2
a
2
+
a
+
1
)
<
f
(
3
a
2
−
4
a
+
1
)
f(2a^2+a+1)<f(3a^2-4a+1)
f
(
2
a
2
+
a
+
1
)
<
f
(
3
a
2
−
4
a
+
1
)
, then the range value of
a
a
a
is________.
7
1
Hide problems
Polynomial
The polynomial
f
(
x
)
=
1
−
x
+
x
2
−
x
3
+
⋯
−
x
19
+
x
20
f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}
f
(
x
)
=
1
−
x
+
x
2
−
x
3
+
⋯
−
x
19
+
x
20
is written into the form
g
(
y
)
=
a
0
+
a
1
y
+
a
2
y
2
+
⋯
+
a
20
y
20
g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}
g
(
y
)
=
a
0
+
a
1
y
+
a
2
y
2
+
⋯
+
a
20
y
20
, where
y
=
x
−
4
y=x-4
y
=
x
−
4
, then
a
0
+
a
1
+
⋯
+
a
20
=
a_0+a_1+\cdots+a_{20}=
a
0
+
a
1
+
⋯
+
a
20
=
________.
6
1
Hide problems
The 2005th Element
Set
T
=
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
,
M
=
{
a
1
7
+
a
2
7
2
+
a
3
7
3
+
a
4
7
4
∣
a
i
∈
T
,
i
=
1
,
2
,
3
,
4
}
T=\{0,1,2,3,4,5,6\},M=\left\{\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4}|a_i\in T,i=1,2,3,4\right\}
T
=
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
,
M
=
{
7
a
1
+
7
2
a
2
+
7
3
a
3
+
7
4
a
4
∣
a
i
∈
T
,
i
=
1
,
2
,
3
,
4
}
. Put all elements in
M
M
M
in order: from small to large, then the 2005th number is
(A)
5
7
+
5
7
2
+
6
7
3
+
3
7
4
\text{(A)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{3}{7^4}
(A)
7
5
+
7
2
5
+
7
3
6
+
7
4
3
(B)
5
7
+
5
7
2
+
6
7
3
+
2
7
4
\text{(B)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{2}{7^4}
(B)
7
5
+
7
2
5
+
7
3
6
+
7
4
2
(C)
1
7
+
1
7
2
+
0
7
3
+
4
7
4
\text{(C)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{4}{7^4}
(C)
7
1
+
7
2
1
+
7
3
0
+
7
4
4
(D)
1
7
+
1
7
2
+
0
7
3
+
3
7
4
\text{(D)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{3}{7^4}
(D)
7
1
+
7
2
1
+
7
3
0
+
7
4
3
5
1
Hide problems
Conics Problem
Which kind of curve does the equation
x
2
sin
2
−
sin
3
+
y
2
cos
2
−
cos
3
=
1
\frac{x^2}{\sin\sqrt2-\sin\sqrt3}+\frac{y^2}{\cos\sqrt2-\cos\sqrt3}=1
s
i
n
2
−
s
i
n
3
x
2
+
c
o
s
2
−
c
o
s
3
y
2
=
1
refer to?
(A)
\text{(A)}
(A)
An ellipse, whose focal points are on
x
x
x
-axis.
(B)
\text{(B)}
(B)
A hyperbola, whose focal points are on
x
x
x
-axis.
(C)
\text{(C)}
(C)
An ellipse, whose focal points are on
y
y
y
-axis.
(D)
\text{(D)}
(D)
A hyperbola, whose focal points are on
y
y
y
-axis.
4
1
Hide problems
3D Geometry
In cube
A
B
C
D
−
A
1
B
1
C
1
D
1
ABCD-A_1B_1C_1D_1
A
BC
D
−
A
1
B
1
C
1
D
1
, draw a plane
α
\alpha
α
perpendicular to line
A
C
′
AC'
A
C
′
, and
α
\alpha
α
has intersections with any surface of the cube. The area of the cross section is
S
S
S
, the perimeter of the cross section is
l
l
l
, then
(A)
\text{(A)}
(A)
The value of
S
S
S
is fixed, but the value of
l
l
l
is not fixed.
(B)
\text{(B)}
(B)
The value of
S
S
S
is not fixed, but the value of
l
l
l
is fixed.
(C)
\text{(C)}
(C)
The value of
S
S
S
is fixed, the value of
l
l
l
is fixed as well.
(D)
\text{(D)}
(D)
The value of
S
S
S
is not fixed, the value of
l
l
l
is not fixed either.
3
2
Hide problems
Geometry
△
A
B
C
\triangle ABC
△
A
BC
is inscribed to unit circle. Bisector of
∠
A
,
∠
B
,
∠
C
\angle A,\angle B,\angle C
∠
A
,
∠
B
,
∠
C
intersect the circle at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
respectively. The value of
A
A
1
⋅
cos
A
2
+
B
B
1
⋅
cos
B
2
+
C
C
1
⋅
cos
C
2
sin
A
+
sin
B
+
sin
C
\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}
s
i
n
A
+
s
i
n
B
+
s
i
n
C
A
A
1
⋅
cos
2
A
+
B
B
1
⋅
cos
2
B
+
C
C
1
⋅
cos
2
C
is
(A)
2
(B)
4
(C)
6
(D)
8
\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8
(A)
2
(B)
4
(C)
6
(D)
8
It Will Take a Lot of Calculation
For positive integer
n
n
n
, define
f
(
n
)
=
{
0
,
if
n
is a perfect square
[
1
{
n
}
]
,
if
n
is not a perfect square
f(n)=\begin{cases} 0, \text{if }n\text{ is a perfect square}\\ \displaystyle \left[\frac{1}{\{\sqrt{n}\}}\right], \text{if }n\text{ is not a perfect square}\\ \end{cases}
f
(
n
)
=
⎩
⎨
⎧
0
,
if
n
is a perfect square
[
{
n
}
1
]
,
if
n
is not a perfect square
. Find the value of
∑
k
=
1
240
f
(
k
)
\sum_{k=1}^{240} f(k)
∑
k
=
1
240
f
(
k
)
. Note:
[
x
]
[x]
[
x
]
is the integral part of real number
x
x
x
, and
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
.
2
2
Hide problems
3D Geometry
Four points in space
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
satisfy that
∣
A
B
∣
=
3
,
∣
B
C
∣
=
7
,
∣
C
D
∣
=
11
,
∣
D
A
∣
=
9
|AB|=3,|BC|=7,|CD|=11,|DA|=9
∣
A
B
∣
=
3
,
∣
BC
∣
=
7
,
∣
C
D
∣
=
11
,
∣
D
A
∣
=
9
, then the number of values of
A
C
→
⋅
B
D
→
\overrightarrow{AC}\cdot\overrightarrow{BD}
A
C
⋅
B
D
is
(A)
\text{(A)}
(A)
Only one.
(B)
\text{(B)}
(B)
Two.
(C)
\text{(C)}
(C)
Three.
(D)
\text{(D)}
(D)
Infinitely many.
Inequality
Positive numbers
a
,
b
,
c
,
x
,
y
,
z
a, b, c, x, y, z
a
,
b
,
c
,
x
,
y
,
z
satisfy that
c
y
+
b
z
=
a
cy + bz = a
cy
+
b
z
=
a
,
a
z
+
c
x
=
b
az + cx = b
a
z
+
c
x
=
b
, and
b
x
+
a
y
=
c
bx + ay = c
b
x
+
a
y
=
c
. Find the minimum value of the function
f
(
x
,
y
,
z
)
=
x
2
x
+
1
+
y
2
y
+
1
+
z
2
z
+
1
f(x,y,z) =\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}
f
(
x
,
y
,
z
)
=
x
+
1
x
2
+
y
+
1
y
2
+
z
+
1
z
2
.
1
2
Hide problems
Inequality
The maximum value of
k
k
k
such that the enequality
x
−
3
+
6
−
x
≥
k
\sqrt{x-3}+\sqrt{6-x}\geq k
x
−
3
+
6
−
x
≥
k
has a real solution is
(A)
6
−
3
(B)
3
(C)
3
+
6
(D)
6
\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6
(A)
6
−
3
(B)
3
(C)
3
+
6
(D)
6
Geometry
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
,
l
l
l
is tangent line of the circumscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
that passes
A
A
A
. The circle with center
A
A
A
and radius
A
C
AC
A
C
, intersects segment
A
B
AB
A
B
at
D
D
D
, and line
l
l
l
at
E
,
F
E, F
E
,
F
(
F
,
B
F,B
F
,
B
are on the same side). Prove that lines
D
E
,
D
F
DE, DF
D
E
,
D
F
pass the incenter and an excenter of
△
A
B
C
\triangle ABC
△
A
BC
respectively.