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Putnam
1996 Putnam
2
Putnam 1996 B2
Putnam 1996 B2
Source:
June 6, 2014
Putnam
inequalities
induction
logarithms
geometry
rectangle
integration
Problem Statement
Prove the inequality for all positive integer
n
n
n
:
(
2
n
−
1
e
)
2
n
−
1
2
<
1
⋅
3
⋅
5
⋯
(
2
n
−
1
)
<
(
2
n
+
1
e
)
2
n
+
1
2
\left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}}
(
e
2
n
−
1
)
2
2
n
−
1
<
1
⋅
3
⋅
5
⋯
(
2
n
−
1
)
<
(
e
2
n
+
1
)
2
2
n
+
1
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