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functions defined by Fibonacci numbers

Source: 16-th Hungary-Israel Binational Mathematical Competition 2005

March 29, 2007
functionalgebra unsolvedalgebra

Problem Statement

Let FnF_{n} be the n−n- th Fibonacci number (where F1=F2=1F_{1}= F_{2}= 1). Consider the functions fn(x)=∥...∥∣x∣−Fn∣−Fn−1∣−...−F2∣−F1∣,gn(x)=∣...∥x−1∣−1∣−...−1∣f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1| (F1+...+FnF_{1}+...+F_{n} one’s). Show that fn(x)=gn(x)f_{n}(x) = g_{n}(x) for every real number x.x.
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