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Putnam
1976 Putnam
1
Putnam 1976 B1
Putnam 1976 B1
Source:
April 18, 2022
college contests
Problem Statement
Evaluate
l
i
m
n
→
∞
1
n
∑
k
=
1
n
(
[
2
n
k
]
−
2
[
n
k
]
)
lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} ([\frac{2n}{k}] -2[\frac{n}{k}])
l
i
m
n
→
∞
n
1
k
=
1
∑
n
([
k
2
n
]
−
2
[
k
n
])
and express your answer in the form
log
a
−
b
,
\log a-b,
lo
g
a
−
b
,
with
a
a
a
and
b
b
b
positive integers. Here
[
x
]
[x]
[
x
]
is defined to be the integer such that
[
x
]
≤
x
<
[
x
]
+
1
[x] \leq x <[x]+1
[
x
]
≤
x
<
[
x
]
+
1
and
log
x
\log x
lo
g
x
is the logarithm of
x
x
x
to base
e
.
e.
e
.
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