Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.
Problem
Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade
Tags: function, algebra, triangle inequality, inequalities