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Functional equation with a) and b)

Source: 2013 Macedonian Additional IMO TST Day 1 P3

February 7, 2016
algebrafunctional equation

Problem Statement

We denote the set of nonzero integers and the set of non-negative integers with Z∗\mathbb Z^* and N0\mathbb N_0, respectively. Find all functions f:Z∗→N0f:\mathbb Z^* \to \mathbb N_0 such that: a)a) f(a+b)≥min(f(a),f(b))f(a+b)\geq min(f(a), f(b)) for all a,ba,b in Z∗\mathbb Z^* for which a+ba+b is in Z∗\mathbb Z^*. b)b) f(ab)=f(a)+f(b)f(ab)=f(a)+f(b) for all a,ba,b in Z∗\mathbb Z^*.
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