MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2008 China Second Round Olympiad
2008 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
a condition for the existence of a sequence
For all
k
=
1
,
2
,
…
,
2008
k=1,2,\ldots,2008
k
=
1
,
2
,
…
,
2008
,
a
k
>
0
a_k>0
a
k
>
0
.Prove that iff
∑
k
=
1
2008
a
k
>
1
\sum_{k=1}^{2008}a_k>1
∑
k
=
1
2008
a
k
>
1
,there exists a function
f
:
N
→
R
f:N\rightarrow R
f
:
N
→
R
satisfying (1)
0
=
f
(
0
)
<
f
(
1
)
<
f
(
2
)
<
…
0=f(0)<f(1)<f(2)<\ldots
0
=
f
(
0
)
<
f
(
1
)
<
f
(
2
)
<
…
; (2)
f
(
n
)
f(n)
f
(
n
)
has a finite limit when
n
n
n
approaches infinity; (3)
f
(
n
)
−
f
(
n
−
1
)
=
∑
k
=
1
2008
a
k
f
(
n
+
k
)
−
∑
k
=
0
2007
a
k
+
1
f
(
n
+
k
)
f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})
f
(
n
)
−
f
(
n
−
1
)
=
∑
k
=
1
2008
a
k
f
(
n
+
k
)
−
∑
k
=
0
2007
a
k
+
1
f
(
n
+
k
)
,for all
n
=
1
,
2
,
3
,
…
n=1,2,3,\ldots
n
=
1
,
2
,
3
,
…
.
2
1
Hide problems
periodic function with certain periods
Let
f
(
x
)
f(x)
f
(
x
)
be a periodic function with periods
T
T
T
and
1
1
1
(
0
<
T
<
1
0<T<1
0
<
T
<
1
).Prove that: (1)If
T
T
T
is rational,then there exists a prime
p
p
p
such that
1
p
\frac{1}{p}
p
1
is also a period of
f
f
f
; (2)If
T
T
T
is irrational,then there exists a strictly decreasing infinite sequence {
a
n
a_n
a
n
},with
1
>
a
n
>
0
1>a_n>0
1
>
a
n
>
0
for all positive integer
n
n
n
,such that all
a
n
a_n
a
n
are periods of
f
f
f
.
1
1
Hide problems
the minimum of f(P) in a quadrilateral
Given a convex quadrilateral with
∠
B
+
∠
D
<
180
\angle B+\angle D<180
∠
B
+
∠
D
<
180
.Let
P
P
P
be an arbitrary point on the plane,define
f
(
P
)
=
P
A
∗
B
C
+
P
D
∗
C
A
+
P
C
∗
A
B
f(P)=PA*BC+PD*CA+PC*AB
f
(
P
)
=
P
A
∗
BC
+
P
D
∗
C
A
+
PC
∗
A
B
. (1)Prove that
P
,
A
,
B
,
C
P,A,B,C
P
,
A
,
B
,
C
are concyclic when
f
(
P
)
f(P)
f
(
P
)
attains its minimum. (2)Suppose that
E
E
E
is a point on the minor arc
A
B
AB
A
B
of the circumcircle
O
O
O
of
A
B
C
ABC
A
BC
,such that
A
E
=
3
2
A
B
,
B
C
=
(
3
−
1
)
E
C
,
∠
E
C
A
=
2
∠
E
C
B
AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB
A
E
=
2
3
A
B
,
BC
=
(
3
−
1
)
EC
,
∠
EC
A
=
2∠
ECB
.Knowing that
D
A
,
D
C
DA,DC
D
A
,
D
C
are tangent to circle
O
O
O
,
A
C
=
2
AC=\sqrt 2
A
C
=
2
,find the minimum of
f
(
P
)
f(P)
f
(
P
)
.