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Fond all functions in M with a) f(1)=5/2, b) f(1)=√3

Source: IMO LongList 1982 - P34

March 18, 2011
functionalgebra proposedalgebra

Problem Statement

Let MM be the set of all functions ff with the following properties:
(i) ff is defined for all real numbers and takes only real values.
(ii) For all x,y∈Rx, y \in \mathbb R the following equality holds: f(x)f(y)=f(x+y)+f(x−y).f(x)f(y) = f(x + y) + f(x - y).
(iii) f(0)≠0.f(0) \neq 0.
Determine all functions f∈Mf \in M such that
(a) f(1)=52f(1)=\frac 52,
(b) f(1)=3f(1)= \sqrt 3.
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