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Functional equation with properties similar to degree map for polynomials

Source: Macedonian TST for IMO 2013 - P3 day 1

March 28, 2021
algebrafunctional equationfunctionnumber theory

Problem Statement

Denote by Z∗\mathbb{Z}^{*} the set of all nonzero integers and denote by N0\mathbb{N}_{0} the set of all nonnegative integers. Find all functions f:Z∗→N0f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0} such that:
(1)(1) For all a,b∈Z∗a,b \in \mathbb{Z}^{*} such that a+b∈Z∗a+b \in \mathbb{Z}^{*} we have f(a+b)≥f(a+b) \geq min {f(a),f(b)}\left \{ f(a),f(b) \right \}.
(2)(2) For all a,b∈Z∗a, b \in \mathbb{Z}^{*} we have f(ab)=f(a)+f(b)f(ab) = f(a)+f(b).
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