Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be real valued function such that $f(0)=0$ and $f(x+y)\geq f(x)+y f(f(x))$. Find all such possible $f$.
When $ x=0$ then $ f(y)\ge 0$ for all $ y\in \mathbb R$ For all $ y\ge 0$ then choose $ x=-y$ we obtain $ f(0)=0\ge f(-y)+ yf(f(-y))$
Source: https://artofproblemsolving.com/community/c6h307863p1662619