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KoMaL A Problems
KoMaL A Problems 2017/2018
A. 709
A. 709
Part of
KoMaL A Problems 2017/2018
Problems
(1)
Minimal C_a so that inequality holds
Source: KöMaL A. 709
12/14/2017
Let
a
>
0
a>0
a
>
0
be a real number. Find the minimal constant
C
a
C_a
C
a
for which the inequality
C
a
∑
k
=
1
n
1
x
k
−
x
k
−
1
>
∑
k
=
1
n
k
+
a
x
k
\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}
C
a
k
=
1
∑
n
x
k
−
x
k
−
1
1
>
k
=
1
∑
n
x
k
k
+
a
holds for any positive integer
n
n
n
and any sequence
0
=
x
0
<
x
1
<
⋯
<
x
n
0=x_0<x_1<\cdots <x_n
0
=
x
0
<
x
1
<
⋯
<
x
n
of real numbers.
inequalities
algebra