View source for 1995 OIM Problems/Problem 6

== 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</math> such that

<cmath>f^n(p) = f( f( \cdots \text{n times} \cdots f(p))) = p</cmath>

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

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

'''Note (*):''' <math>\left\lfloor x \right\rfloor</math> indicates the largest integer less than or equal to <math>x</math>.

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

== Solution ==
{{solution}}

== See also ==
https://www.oma.org.ar/enunciados/ibe10.htm