IMO 1979 LL NET49

Origin: NET

Problem

Let there be given two sequences of integers fi(1), fi(2), . . . (i = 1, 2) satisfying: (i) fi(nm) = fi(n)fi(m) if gcd(n, m) = 1; (ii) for every prime P and all k = 2, 3, 4, . . ., fi(P k) = fi(P)fi(P k−1) − P 2f(P k−2). Moreover, for every prime P: (iii) f1(P) = 2P, (iv) f2(P) < 2P. Prove that |f2(n)| < f1(n) for all n.