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Tow Problems - Printable Version

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Tow Problems - elim - 06-01-2010 10:52 PM

1. Let $g(n)$ be the greatest odd divisor of $n$, show that $
\mathop {\lim }\limits_{n \to + \infty } \frac{1}
{n} \cdot \sum\limits_{k = 1}^n {\frac{{g\left( k \right)}}
{k}}
$ exists and find it ( Bulgaria 1985)
2. Find all $
k \in \mathbb{Z}\text{ with }k \geq 2
$ such that $
n \not|g(k^n+1)
$, $
\forall n \in \mathbb{Z}, \quad (n>1)
$ (OlimpĂ­ada Rioplatense 2008) - $g(n)$ is defined as in 1 -