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Germany Contests
German National Olympiad
1997 German National Olympiad
2
2
Part of
1997 German National Olympiad
Problems
(1)
1/2^n \sum_{k = 1}^{2^n} \frac{u(k)}{k}> 2/3, greatest odd divisor
Source: Germany 1997 p2
2/22/2020
For a positive integer
k
k
k
, let us denote by
u
(
k
)
u(k)
u
(
k
)
the greatest odd divisor of
k
k
k
. Prove that, for each
n
∈
N
n \in N
n
∈
N
,
1
2
n
∑
k
=
1
2
n
u
(
k
)
k
>
2
3
\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}
2
n
1
∑
k
=
1
2
n
k
u
(
k
)
>
3
2
.
inequalities
greatest odd divisor
divisor
number theory