Problem

Source: Iran TST 2012-First exam-2nd day-P5

Tags: function, induction, ceiling function, binomial coefficients, algebra proposed, algebra



The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: a) $f(a)=0 \Leftrightarrow a=0$ b) $f(ab)=f(a)f(b)$ c) $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. Proposed by Masoud Shafaei