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Ireland Contests
Ireland National Math Olympiad
1996 Irish Math Olympiad
2
inequality with integers
inequality with integers
Source: Ireland 1996
July 1, 2009
inequalities
inequalities proposed
Problem Statement
Show that for every positive integer
n
n
n
,
2
1
2
⋅
4
1
4
⋅
8
1
8
⋅
.
.
.
⋅
(
2
n
)
1
2
n
<
4
2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4
2
2
1
⋅
4
4
1
⋅
8
8
1
⋅
...
⋅
(
2
n
)
2
n
1
<
4
.
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