IMO 1982 LL GDR34

Let M be the set of all functions f with the following proper-

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IMO 1982 LL GDR34

Origin: GDR

Problem

Let M be the set of all functions f with the following proper- ties: (i) f is defined for all real numbers and takes only real values. (ii) For all x, y \inR the following equality holds: f(x)f(y) = f(x + y) + f(x −y). (iii) f(0) ̸= 0. Determine all functions f \inM such that (a) f(1) = 5/2; (b) f(1) = \sqrt 3.