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Putnam
2015 Putnam
A4
A4
Part of
2015 Putnam
Problems
(1)
Putnam 2015 A4
Source:
12/6/2015
For each real number
x
,
x,
x
,
let
f
(
x
)
=
∑
n
∈
S
x
1
2
n
f(x)=\sum_{n\in S_x}\frac1{2^n}
f
(
x
)
=
n
∈
S
x
∑
2
n
1
where
S
x
S_x
S
x
is the set of positive integers
n
n
n
for which
⌊
n
x
⌋
\lfloor nx\rfloor
⌊
n
x
⌋
is even.What is the largest real number
L
L
L
such that
f
(
x
)
≥
L
f(x)\ge L
f
(
x
)
≥
L
for all
x
∈
[
0
,
1
)
x\in [0,1)
x
∈
[
0
,
1
)
?(As usual,
⌊
z
⌋
\lfloor z\rfloor
⌊
z
⌋
denotes the greatest integer less than or equal to
z
.
z.
z
.
Putnam
Putnam 2015