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Difference between revisions of "1995 OIM Problems/Problem 6"

(Created page with "== Problem == A function <math>f: N \to N</math> is circular if for every <math>p</math> in <math>N</math> there exists <math>n</math> in <math>N</math> with <math>n \le p</ma...")
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Revision as of 14:56, 13 December 2023

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

Solution

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See also

https://www.oma.org.ar/enunciados/ibe10.htm

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