1995 OIM Problems/Problem 6

Problem

A function $f: N \to N$ is circular if for every $p$ in $N$ there exists $n$ in $N$ with $n \le p$ such that

$f^n(p) = f( f( \cdots n \; times \cdots f(p))) = p$

The function f has degree of repulsion $k$ , $0 < k<1$ , if for each $p$ in $N$ , $f^i(p) \ne p$ for $i=1, 2, \cdots , \left\lfloor k.p \right\rfloor$ (*).

Find the greatest degree of repulsion that a circular function can have.

Note (*): $\left\lfloor x \right\rfloor$ indicates the largest integer less than or equal to $x$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1995 OIM Problems/Problem 6

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