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Find all functions

Source: Vietnam NMO 1985 Problem 2

February 12, 2009

functionalgebra unsolvedalgebra

Problem Statement

Find all functions

f \colon \mathbb{Z} \mapsto \mathbb{R}

which satisfy: i) f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y) for all integers

x

y

ii)

f(0) \neq 0

iii) f(1) \equal{} \frac {5}{2}