MathDB
Problems
Contests
National and Regional Contests
China Contests
National High School Mathematics League
2002 National High School Mathematics League
2002 National High School Mathematics League
Part of
National High School Mathematics League
Subcontests
(15)
15
1
Hide problems
Quadratic Function
Quadratic function
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=ax^2+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
satisfies that:
(
1
)
∀
x
∈
R
,
f
(
x
−
4
)
=
f
(
2
−
x
)
,
f
(
x
)
≥
x
(1)\forall x\in\mathbb{R},f(x-4)=f(2-x),f(x)\geq x
(
1
)
∀
x
∈
R
,
f
(
x
−
4
)
=
f
(
2
−
x
)
,
f
(
x
)
≥
x
;
(
2
)
∀
x
∈
(
0
,
2
)
,
f
(
x
)
≤
(
x
+
1
2
)
2
(2)\forall x\in(0,2),f(x)\leq\left(\frac{x+1}{2}\right)^2
(
2
)
∀
x
∈
(
0
,
2
)
,
f
(
x
)
≤
(
2
x
+
1
)
2
;
(
3
)
min
x
∈
R
f
(
x
)
=
0
(3)\min\limits_{x\in\mathbb{R}}f(x)=0
(
3
)
x
∈
R
min
f
(
x
)
=
0
Find the maximum of
m
(
m
>
1
)
m(m>1)
m
(
m
>
1
)
, satisfying: There exists
t
∈
R
t\in\mathbb{R}
t
∈
R
, as long as
x
∈
[
1
,
m
]
x\in[1,m]
x
∈
[
1
,
m
]
, then
f
(
x
+
t
)
≤
x
f(x+t)\leq x
f
(
x
+
t
)
≤
x
.
14
1
Hide problems
A Lot of Curves
There is a family of curves:
P
0
,
P
1
,
P
2
,
⋯
P_0,P_1,P_2,\cdots
P
0
,
P
1
,
P
2
,
⋯
.
P
0
P_0
P
0
is a regular triangle, whose area is
1
1
1
. For all
k
∈
Z
+
k\in\mathbb{Z}_+
k
∈
Z
+
,
P
k
P_k
P
k
is defined in this way: trisect all sides of
P
k
−
1
P_{k-1}
P
k
−
1
, and draw outward a regular triangle with side of the segment in the middle, then cut off the segment in the middle.
S
n
S_n
S
n
is the area of
P
n
P_n
P
n
. (a) Find
S
n
S_n
S
n
. (b) Find
lim
n
→
∞
S
n
\lim_{n\to\infty}S_n
lim
n
→
∞
S
n
.
13
1
Hide problems
Parabola
A
(
0
,
2
)
A(0,2)
A
(
0
,
2
)
, and two points
B
,
C
B,C
B
,
C
on parabola
y
2
=
x
+
4
y^2=x+4
y
2
=
x
+
4
satisfy that
A
B
⊥
B
C
AB\perp BC
A
B
⊥
BC
. Find the range value of
y
C
y_C
y
C
.
12
1
Hide problems
Trigonometry
For all
x
∈
R
x\in\mathbb{R}
x
∈
R
,
sin
2
x
+
a
cos
x
+
a
2
≥
1
+
cos
x
\sin^2 x+a\cos x+a^2\geq 1+\cos x
sin
2
x
+
a
cos
x
+
a
2
≥
1
+
cos
x
, then the range value of negative number
a
a
a
is________
11
1
Hide problems
The Minumum Value
If
log
4
(
x
+
2
y
)
+
log
4
(
x
−
2
y
)
=
1
\log_4 (x+2y)+\log_4 (x-2y)=1
lo
g
4
(
x
+
2
y
)
+
lo
g
4
(
x
−
2
y
)
=
1
, then the minumum value of
∣
x
∣
−
∣
y
∣
|x|-|y|
∣
x
∣
−
∣
y
∣
is________.
10
1
Hide problems
A Function
f
(
x
)
f(x)
f
(
x
)
is a function defined on
R
\mathbb{R}
R
.
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
, and for all
x
∈
R
x\in\mathbb{R}
x
∈
R
,
f
(
x
+
5
)
≥
x
+
5
,
f
(
x
+
1
)
≤
f
(
x
)
+
1
f(x+5)\geq x+5,f(x+1)\leq f(x)+1
f
(
x
+
5
)
≥
x
+
5
,
f
(
x
+
1
)
≤
f
(
x
)
+
1
. If
g
(
x
)
=
f
(
x
)
+
1
−
x
g(x)=f(x)+1-x
g
(
x
)
=
f
(
x
)
+
1
−
x
, then
g
(
2002
)
=
g(2002)=
g
(
2002
)
=
________.
9
1
Hide problems
Ten Points
Points
P
1
,
P
2
,
P
3
,
P
4
P_1,P_2,P_3,P_4
P
1
,
P
2
,
P
3
,
P
4
are vertexes of a regular triangular pyramid, and
P
5
,
P
6
,
P
7
,
P
8
,
P
9
,
P
10
P_5,P_6,P_7,P_8,P_9,P_{10}
P
5
,
P
6
,
P
7
,
P
8
,
P
9
,
P
10
midpoints of edges. The number of groups
(
P
1
,
P
i
,
P
j
,
P
k
)
(
1
<
i
<
j
<
k
≤
10
)
(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)
(
P
1
,
P
i
,
P
j
,
P
k
)
(
1
<
i
<
j
<
k
≤
10
)
that
P
1
,
P
i
,
P
j
,
P
k
P_1,P_i,P_j,P_k
P
1
,
P
i
,
P
j
,
P
k
are coplane is________.
8
1
Hide problems
Binomial Theorem
Consider the expanded form of
(
x
+
1
2
x
4
)
n
\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n
(
x
+
2
4
x
1
)
n
, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.
7
1
Hide problems
Complex Number
Complex numbers
∣
z
1
∣
=
2
,
∣
z
2
∣
=
3
|z_1|=2,|z_2|=3
∣
z
1
∣
=
2
,
∣
z
2
∣
=
3
, and the intersection angle between the vectors corresponding to
z
1
,
z
2
z_1,z_2
z
1
,
z
2
is
6
0
∘
60^{\circ}
6
0
∘
, then
∣
z
1
+
z
2
∣
∣
z
1
−
z
2
∣
=
\frac{|z_1+z_2|}{|z_1-z_2|}=
∣
z
1
−
z
2
∣
∣
z
1
+
z
2
∣
=
________.
6
1
Hide problems
Geometry Volume
Consider the area encircled by
x
2
=
4
y
,
x
2
=
−
4
y
,
x
=
4
,
x
=
−
4
x^2=4y,x^2=-4y,x=4,x=-4
x
2
=
4
y
,
x
2
=
−
4
y
,
x
=
4
,
x
=
−
4
, rotate it around
y
y
y
-axis, the volume of the revolved body is
V
1
V_1
V
1
. Then consider the figure formed by all points
(
x
,
y
)
(x,y)
(
x
,
y
)
that
x
2
+
y
2
≤
16
,
x
2
+
(
y
−
2
)
2
≥
4
,
x
2
+
(
y
−
2
)
2
≥
4
x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4
x
2
+
y
2
≤
16
,
x
2
+
(
y
−
2
)
2
≥
4
,
x
2
+
(
y
−
2
)
2
≥
4
, rotate it around
y
y
y
-axis, the volume of the revolved body is
V
2
V_2
V
2
. The relationship between
V
1
V_1
V
1
and
V
2
V_2
V
2
is
(A)
V
1
=
1
2
V
2
(B)
V
1
=
2
3
V
2
(C)
V
1
=
V
2
(D)
V
1
=
2
V
2
\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2
(A)
V
1
=
2
1
V
2
(B)
V
1
=
3
2
V
2
(C)
V
1
=
V
2
(D)
V
1
=
2
V
2
5
1
Hide problems
Mapping Problem
Two sets of real numbers
A
=
{
a
1
,
a
2
,
⋯
,
a
100
}
,
B
=
{
b
1
,
b
2
,
⋯
,
b
50
}
A=\{a_1,a_2,\cdots,a_{100}\},B=\{b_1,b_2,\cdots,b_{50}\}
A
=
{
a
1
,
a
2
,
⋯
,
a
100
}
,
B
=
{
b
1
,
b
2
,
⋯
,
b
50
}
. Mapping
f
:
A
→
B
f:A\to B
f
:
A
→
B
,
∀
i
(
1
≤
i
≤
50
)
,
∃
j
(
1
≤
j
≤
100
)
,
f
(
a
j
)
=
b
i
\forall i(1\leq i\leq 50),\exists j(1\leq j\leq100),f(a_j)=b_i
∀
i
(
1
≤
i
≤
50
)
,
∃
j
(
1
≤
j
≤
100
)
,
f
(
a
j
)
=
b
i
, and
f
(
a
1
)
≤
f
(
a
2
)
≤
⋯
≤
f
(
a
100
)
f(a_1)\leq f(a_2)\leq\cdots\leq f(a_{100})
f
(
a
1
)
≤
f
(
a
2
)
≤
⋯
≤
f
(
a
100
)
Then the number of different
f
f
f
is
(A)
C
100
50
(B)
C
99
50
(C)
C
100
49
(D)
C
99
49
\text{(A)}\text{C}_{100}^{50}\qquad\text{(B)}\text{C}_{99}^{50}\qquad\text{(C)}\text{C}_{100}^{49}\qquad\text{(D)}\text{C}_{99}^{49}
(A)
C
100
50
(B)
C
99
50
(C)
C
100
49
(D)
C
99
49
4
1
Hide problems
Line And Ellipse
Line
x
4
+
y
3
=
1
\frac{x}{4}+\frac{y}{3}=1
4
x
+
3
y
=
1
and ellipse
x
2
16
+
y
2
9
=
1
\frac{x^2}{16}+\frac{y^2}{9}=1
16
x
2
+
9
y
2
=
1
intersect at
A
A
A
and
B
B
B
. A point on the ellipse
P
P
P
satisties that the area of
△
P
A
B
\triangle PAB
△
P
A
B
is
3
3
3
. The number of such points is
(A)
1
(B)
2
(C)
3
(D)
4
\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4
(A)
1
(B)
2
(C)
3
(D)
4
3
2
Hide problems
A Function
Function
f
(
x
)
=
x
1
−
2
x
−
x
2
f(x)=\frac{x}{1-2^x}-\frac{x}{2}
f
(
x
)
=
1
−
2
x
x
−
2
x
is
(A)
\text{(A)}
(A)
an even function, not an odd function.
(B)
\text{(B)}
(B)
an odd function, not an even function.
(C)
\text{(C)}
(C)
an even function, also an odd function.
(D)
\text{(D)}
(D)
neither an even function, nor an odd function.
The FIFA World Cup
Before the FIFA world cup, the football coach of F country want to test seven players
A
1
,
A
2
,
⋯
,
A
7
A_1, A_2, \cdots, A_7
A
1
,
A
2
,
⋯
,
A
7
. He asks them to join in three training matches (90 minutes each), and everyone must appear in each match at least once. Suppose that at any moment during a match, one and only one of them enters the field, and the total time (measured in minutes) on the field for
A
1
,
A
2
,
A
3
,
A
4
A_1, A_2, A_3, A_4
A
1
,
A
2
,
A
3
,
A
4
are multiples of
7
7
7
and the total time for
A
5
,
A
6
,
A
7
A_5, A_6, A_7
A
5
,
A
6
,
A
7
are multiples of
13
13
13
. If the number of substitutions of players during each match is not limited, find the number of different cases. Note: If and only if the total time of a certian player is different, then the case is considered different.
2
2
Hide problems
The Minumum Value
Real numbers
x
,
y
x,y
x
,
y
satisfy that
(
x
+
5
)
2
+
(
y
−
12
)
2
=
1
4
2
(x+5)^2+(y-12)^2=14^2
(
x
+
5
)
2
+
(
y
−
12
)
2
=
1
4
2
, then the minumum value of
x
2
+
y
2
x^2+y^2
x
2
+
y
2
is
(A)
2
(B)
1
(C)
3
(D)
2
\text{(A)}2\qquad\text{(B)}1\qquad\text{(C)}\sqrt3\qquad\text{(D)}\sqrt2\qquad
(A)
2
(B)
1
(C)
3
(D)
2
Cubic Equation
For real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
and positive number
λ
\lambda
λ
such that three real roots
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
of
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f(x)=x^3+ax^2+bx+c
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
satisfying:
(
1
)
x
2
−
x
1
=
λ
(1) x_2-x_1=\lambda
(
1
)
x
2
−
x
1
=
λ
;
(
2
)
x
3
>
1
2
(
x
1
+
x
2
)
(2) x_3>\frac{1}{2}(x_1+x_2)
(
2
)
x
3
>
2
1
(
x
1
+
x
2
)
. Find the maximum value of
2
a
3
+
27
c
−
9
a
b
λ
3
\frac{2a^3+27c-9ab}{\lambda^3}
λ
3
2
a
3
+
27
c
−
9
ab
1
2
Hide problems
A Function
The increasing interval of
f
(
x
)
=
log
1
2
(
x
2
−
2
x
−
3
)
f(x)=\log_{\frac{1}{2}}(x^2-2x-3)
f
(
x
)
=
lo
g
2
1
(
x
2
−
2
x
−
3
)
is
(A)
(
−
∞
,
−
1
)
(B)
(
−
∞
,
1
)
(C)
(
1
,
+
∞
)
(D)
(
3
,
+
∞
)
\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)
(A)
(
−
∞
,
−
1
)
(B)
(
−
∞
,
1
)
(C)
(
1
,
+
∞
)
(D)
(
3
,
+
∞
)
Geometry
In
△
A
B
C
\triangle ABC
△
A
BC
,
∠
A
=
6
0
∘
\angle A = 60^{\circ}
∠
A
=
6
0
∘
,
A
B
>
A
C
AB>AC
A
B
>
A
C
,
O
O
O
is the circumcenter and
H
H
H
is the intersection point of two heights
B
E
BE
BE
and
C
F
CF
CF
. Points
M
M
M
and
N
N
N
lie on segments
B
H
BH
B
H
and
H
F
HF
H
F
respectively, and
B
M
=
C
N
BM=CN
BM
=
CN
. Find the value of
M
H
+
N
H
O
H
\frac{MH+NH}{OH}
O
H
M
H
+
N
H
.