MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1984 Miklós Schweitzer
5
Miklós Schweitzer 1984- Problem 5
Miklós Schweitzer 1984- Problem 5
Source:
September 4, 2016
college contests
Problem Statement
5. Let
a
0
,
a
1
,
…
a_0 , a_1 , \dots
a
0
,
a
1
,
…
be nonnegative real numbers such that
∑
n
=
0
∞
a
n
=
∞
\sum_{n=0}^{\infty}a_n = \infty
∑
n
=
0
∞
a
n
=
∞
For arbitrary
c
>
0
c>0
c
>
0
, let
n
j
(
c
)
=
min
{
k
:
c
.
j
≤
∑
i
=
0
k
a
i
}
n_{j}(c)= \min \left \{ k : c.j \leq \sum_{i=0}^{k} a_i \right \}
n
j
(
c
)
=
min
{
k
:
c
.
j
≤
∑
i
=
0
k
a
i
}
,
j
=
1
,
2
,
…
j= 1,2, \dots
j
=
1
,
2
,
…
Prove that if
∑
i
=
0
∞
a
i
2
=
∞
\sum_{i=0}^{\infty}a_i^2 = \infty
∑
i
=
0
∞
a
i
2
=
∞
, then there exists a
c
>
0
c>0
c
>
0
for which
∑
j
=
1
∞
a
n
j
(
c
)
=
∞
\sum_{j=1}^{\infty} a_{n_j (c)} = \infty
∑
j
=
1
∞
a
n
j
(
c
)
=
∞
.(S.24) [P. Erdos, I. Joó, L. Székely]
Back to Problems
View on AoPS