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IMO Shortlist 1966 problem 30

1966 shortlist
Let n be a positive integer, prove that :

(a) \log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;

(b) \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).

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1225IMO Shortlist 1966 problem 42
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