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f(a) = f(1995),..., f(n + a) = (f(n) - 1)/(f(n) + 1)

Source: Nordic Mathematical Contest 1996 #4

October 4, 2017
algebrafunctional equation in N

Problem Statement

The real-valued function ff is defined for positive integers, and the positive integer aa satisfies f(a)=f(1995),f(a+1)=f(1996),f(a+2)=f(1997),f(n+a)=f(n)−1f(n)+1f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1} for all positive integers nn. (i) Show that f(n+4a)=f(n)f(n+ 4a) = f(n) for all positive integers nn. (ii) Determine the smallest possible aa.
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