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Iterating functions

Source: IMOTC 2014 Practice Test 2 Problem 3

July 11, 2014
functionnumber theory unsolvednumber theory

Problem Statement

For integers a,ba,b we define f((a,b))=(2a,b−a)f((a,b))=(2a,b-a) if a<ba<b and f((a,b))=(a−b,2b)f((a,b))=(a-b,2b) if a≥ba\geq b. Given a natural number n>1n>1 show that there exist natural numbers m,km,k with m<nm<n such that fk((n,m))=(m,n)f^{k}((n,m))=(m,n),where fk(x)=f(f(f(...f(x))))f^{k}(x)=f(f(f(...f(x)))),ff being composed with itself kk times.
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