Problem

Source: Iran MO 3rd round midterm exam

Tags: functional equation, Iran



Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ : $$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. Proposed by Amirhossein Zolfaghari