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10th ibmo - chile 1995/q6.

Source: Spanish Communities

May 7, 2006
functionfloor functioninductioncalculusintegrationalgebra unsolvedalgebra

Problem Statement

A function f:N→Nf: \N\rightarrow\N is circular if for every p∈Np\in\N there exists n∈N, n≤pn\in\N,\ n\leq{p} such that fn(p)=pf^n(p)=p (ff composed with itself nn times) The function ff has repulsion degree k>0k>0 if for every p∈Np\in\N fi(p)≠pf^i(p)\neq{p} for every i=1,2,…,⌊kp⌋i=1,2,\dots,\lfloor{kp}\rfloor. Determine the maximum repulsion degree can have a circular function.
Note: Here ⌊x⌋\lfloor{x}\rfloor is the integer part of xx.
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