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f(a) + f(b)<= f (a + b) <= f(a) + f(b) + 1 if kf(n) <= f (kn) <= kf(n)+ k- 1

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 C2 p5

September 23, 2021
algebrainequalitiesFunctional inequalityfunctional

Problem Statement

Let f:N+→N+f: N^+ \to N^+ be a function that satisfies that kf(n)≤f(kn)≤kf(n)+k−1,  ∀k,n∈N+kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+ Prove that f(a)+f(b)≤f(a+b)≤f(a)+f(b)+1,  ∀a,b∈N+f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+
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