MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2011 China Second Round Olympiad
2011 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(11)
11
1
Hide problems
Ellipse
A line
ℓ
\ell
ℓ
with slope of
1
3
\frac{1}{3}
3
1
insects the ellipse
C
:
x
2
36
+
y
2
4
=
1
C:\frac{x^2}{36}+\frac{y^2}{4}=1
C
:
36
x
2
+
4
y
2
=
1
at points
A
,
B
A,B
A
,
B
and the point
P
(
3
2
,
2
)
P\left( 3\sqrt{2} , \sqrt{2}\right)
P
(
3
2
,
2
)
is above the line
ℓ
\ell
ℓ
. (1) Prove that the locus of the incenter of triangle
P
A
B
PAB
P
A
B
is a segment,(2) If
∠
A
P
B
=
π
3
\angle APB=\frac{\pi}{3}
∠
A
PB
=
3
π
, then find the area of triangle
P
A
B
PAB
P
A
B
.
10
1
Hide problems
sequence a_n
A sequence
a
n
a_n
a
n
satisfies
a
1
=
2
t
−
3
a_1 =2t-3
a
1
=
2
t
−
3
(
t
≠
1
,
−
1
t \ne 1,-1
t
=
1
,
−
1
), and
a
n
+
1
=
(
2
t
n
+
1
−
3
)
a
n
+
2
(
t
−
1
)
t
n
−
1
a
n
+
2
t
n
−
1
a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}
a
n
+
1
=
a
n
+
2
t
n
−
1
(
2
t
n
+
1
−
3
)
a
n
+
2
(
t
−
1
)
t
n
−
1
.i) Find
a
n
a_n
a
n
,ii) If
t
>
0
t>0
t
>
0
, compare
a
n
+
1
a_{n+1}
a
n
+
1
with
a
n
a_n
a
n
.
9
1
Hide problems
functionproblem
Let
f
(
x
)
=
∣
log
(
x
+
1
)
∣
f(x)=|\log(x+1)|
f
(
x
)
=
∣
lo
g
(
x
+
1
)
∣
and let
a
,
b
a,b
a
,
b
be two real numbers (
a
<
b
a<b
a
<
b
) satisfying the equations
f
(
a
)
=
f
(
−
b
+
1
a
+
1
)
f(a)=f\left(-\frac{b+1}{a+1}\right)
f
(
a
)
=
f
(
−
a
+
1
b
+
1
)
and
f
(
10
a
+
6
b
+
21
)
=
4
log
2
f\left(10a+6b+21\right)=4\log 2
f
(
10
a
+
6
b
+
21
)
=
4
lo
g
2
. Find
a
,
b
a,b
a
,
b
.
8
1
Hide problems
integers in the sequence
Given that
a
n
=
(
200
n
)
⋅
6
200
−
n
3
⋅
(
1
2
)
n
a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n
a
n
=
(
n
200
)
⋅
6
3
200
−
n
⋅
(
2
1
)
n
(
1
≤
n
≤
95
1 \leq n \leq 95
1
≤
n
≤
95
). How many integers are there in the sequence
{
a
n
}
\{a_n\}
{
a
n
}
?
7
1
Hide problems
parabola
The line
x
−
2
y
−
1
=
0
x-2y-1=0
x
−
2
y
−
1
=
0
insects the parabola
y
2
=
4
x
y^2=4x
y
2
=
4
x
at two different points
A
,
B
A, B
A
,
B
. Let
C
C
C
be a point on the parabola such that
∠
A
C
B
=
π
2
\angle ACB=\frac{\pi}{2}
∠
A
CB
=
2
π
. Find the coordinate of point
C
C
C
.
6
1
Hide problems
radius of a circumsphere
In a tetrahedral
A
B
C
D
ABCD
A
BC
D
, given that
∠
A
D
B
=
∠
B
D
C
=
∠
C
D
A
=
π
3
\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}
∠
A
D
B
=
∠
B
D
C
=
∠
C
D
A
=
3
π
,
A
D
=
B
D
=
3
AD=BD=3
A
D
=
B
D
=
3
, and
C
D
=
2
CD=2
C
D
=
2
. Find the radius of the circumsphere of
A
B
C
D
ABCD
A
BC
D
.
5
1
Hide problems
how many kinds of arrangements are there
We want to arrange
7
7
7
students to attend
5
5
5
sports events, but students
A
A
A
and
B
B
B
can't take part in the same event, every event has its own participants, and every student can only attend one event. How many arrangements are there?
4
2
Hide problems
matrix and elements
Let
A
A
A
be a
3
×
9
3 \times 9
3
×
9
matrix. All elements of
A
A
A
are positive integers. We call an
m
×
n
m\times n
m
×
n
submatrix of
A
A
A
"ox" if the sum of its elements is divisible by
10
10
10
, and we call an element of
A
A
A
"carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in
A
A
A
.
trigonometry
If
cos
5
x
−
sin
5
x
<
7
(
sin
3
x
−
cos
3
x
)
{\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x})
cos
5
x
−
sin
5
x
<
7
(
sin
3
x
−
cos
3
x
)
(for
x
∈
[
0
,
2
π
)
x\in [ 0,2\pi)
x
∈
[
0
,
2
π
)
), then find the range of
x
x
x
.
2
2
Hide problems
a polynomial
For any integer
n
≥
4
n\ge 4
n
≥
4
, prove that there exists a
n
n
n
-degree polynomial
f
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0
f
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
satisfying the two following properties:(1)
a
i
a_i
a
i
is a positive integer for any
i
=
0
,
1
,
…
,
n
−
1
i=0,1,\ldots,n-1
i
=
0
,
1
,
…
,
n
−
1
, and(2) For any two positive integers
m
m
m
and
k
k
k
(
k
≥
2
k\ge 2
k
≥
2
) there exist distinct positive integers
r
1
,
r
2
,
.
.
.
,
r
k
r_1,r_2,...,r_k
r
1
,
r
2
,
...
,
r
k
, such that
f
(
m
)
≠
∏
i
=
1
k
f
(
r
i
)
f(m)\ne\prod_{i=1}^{k}f(r_i)
f
(
m
)
=
∏
i
=
1
k
f
(
r
i
)
.
Range of a function
Find the range of the function
f
(
x
)
=
x
2
+
1
x
−
1
f(x)=\frac{\sqrt{x^2+1}}{x-1}
f
(
x
)
=
x
−
1
x
2
+
1
.
1
2
Hide problems
two angles are equal
Let
P
,
Q
P,Q
P
,
Q
be the midpoints of diagonals
A
C
,
B
D
AC,BD
A
C
,
B
D
in cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. If
∠
B
P
A
=
∠
D
P
A
\angle BPA=\angle DPA
∠
BP
A
=
∠
D
P
A
, prove that
∠
A
Q
B
=
∠
C
Q
B
\angle AQB=\angle CQB
∠
A
QB
=
∠
CQB
.
Set
Let the set
A
=
(
a
1
,
a
2
,
a
3
,
a
4
)
A=(a_{1},a_{2},a_{3},a_{4})
A
=
(
a
1
,
a
2
,
a
3
,
a
4
)
. If the sum of elements in every 3-element subset of
A
A
A
makes up the set
B
=
(
−
1
,
5
,
3
,
8
)
B=(-1,5,3,8)
B
=
(
−
1
,
5
,
3
,
8
)
, then find the set
A
A
A
.
3
2
Hide problems
maximum of {i,j,k}
Given
n
≥
4
n\ge 4
n
≥
4
real numbers
a
n
>
.
.
.
>
a
1
>
0
a_{n}>...>a_{1} > 0
a
n
>
...
>
a
1
>
0
. For
r
>
0
r > 0
r
>
0
, let
f
n
(
r
)
f_{n}(r)
f
n
(
r
)
be the number of triples
(
i
,
j
,
k
)
(i,j,k)
(
i
,
j
,
k
)
with
1
≤
i
<
j
<
k
≤
n
1\leq i<j<k\leq n
1
≤
i
<
j
<
k
≤
n
such that
a
j
−
a
i
a
k
−
a
j
=
r
\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r
a
k
−
a
j
a
j
−
a
i
=
r
. Prove that
f
n
(
r
)
<
n
2
4
{f_{n}(r)}<\frac{n^{2}}{4}
f
n
(
r
)
<
4
n
2
.
find log-a b
Let
a
,
b
a,b
a
,
b
be positive reals such that
1
a
+
1
b
≤
2
2
\frac{1}{a}+\frac{1}{b}\leq2\sqrt2
a
1
+
b
1
≤
2
2
and
(
a
−
b
)
2
=
4
(
a
b
)
3
(a-b)^2=4(ab)^3
(
a
−
b
)
2
=
4
(
ab
)
3
. Find
log
a
b
\log_a b
lo
g
a
b
.