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Find all pairs of functions

Source: South African MO 2008 Q6

May 27, 2012
functiontrigonometryquadraticsalgebra unsolvedalgebra

Problem Statement

Find all function pairs (f,g)(f,g) where each ff and gg is a function defined on the integers and with values, such that, for all integers aa and bb, f(a+b)=f(a)g(b)+g(a)f(b)g(a+b)=g(a)g(b)āˆ’f(a)f(b).f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).
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