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P 6 Functional equation - 27th Indian NMO 2012

Source: INMO

February 5, 2012
functiongroup theoryalgebra proposedalgebrafunctional equation

Problem Statement

Let f:Z→Zf : \mathbb{Z} \to \mathbb{Z} be a function satisfying f(0)≠0f(0) \ne 0, f(1)=0f(1) = 0 and
(i)f(xy)+f(x)f(y)=f(x)+f(y)(i) f(xy) + f(x)f(y) = f(x) + f(y)
(ii)(f(x−y)−f(0))f(x)f(y)=0(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0
for all x,y∈Zx,y \in \mathbb{Z}, simultaneously.
(a)(a) Find the set of all possible values of the function ff.
(b)(b) If f(10)≠0f(10) \ne 0 and f(2)=0f(2) = 0, find the set of all integers nn such that f(n)≠0f(n) \ne 0.
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