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1966 IMO Shortlist
30
Logarithmic inequalities
Logarithmic inequalities
Source:
September 28, 2010
inequalities
logarithms
calculus
IMO Shortlist
IMO Longlist
Problem Statement
Let
n
n
n
be a positive integer, prove that :(a)
log
10
(
n
+
1
)
>
3
10
n
+
log
10
n
;
\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;
lo
g
10
(
n
+
1
)
>
10
n
3
+
lo
g
10
n
;
(b)
log
n
!
>
3
n
10
(
1
2
+
1
3
+
⋯
+
1
n
−
1
)
.
\log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).
lo
g
n
!
>
10
3
n
(
2
1
+
3
1
+
⋯
+
n
1
−
1
)
.
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