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Extremely Similar to ISL 2014 A4

Source: 2018 Thailand October Camp 1.3

February 18, 2022
functional equationalgebrafunction

Problem Statement

Find all function f:Z→Zf:\mathbb{Z}\to\mathbb{Z} satisfying
(i)\text{(i)} f(f(m)+n)+2m=f(n)+f(3m)f(f(m)+n)+2m=f(n)+f(3m) for every m,n∈Zm,n\in\mathbb{Z},
(ii)\text{(ii)} there exists a d∈Zd\in\mathbb{Z} such that f(d)−f(0)=2f(d)-f(0)=2, and
(iii)\text{(iii)} f(1)−f(0)f(1)-f(0) is even.
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