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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2008 China Second Round Olympiad
3
a condition for the existence of a sequence
a condition for the existence of a sequence
Source: China second round 2008 p3
March 3, 2012
function
algebra unsolved
algebra
Problem Statement
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
,
…
.
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