(IDMO 3, Problem 6) Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ such that the set of functions $g:\mathbb R\to\mathbb R$ satisfying $$g(a)f(b)+g(b)f(a)\leqslant (a+f(a))(b+f(b)) \text{ for all }a,b\in\mathbb R$$is finite but nonempty.

Q. (Caro-Wei) Given a graph $G$, there exists an independent set (i.e. no two vertices are adjacent) of size at least $$\sum_{i=1}^{|V|} \frac{1}{\deg v + 1}$$

I took part in the InfinityDots MO hosted by talkon.

Here are some problems which build towards extremal constructions using some kind of “dynamic” argument.

This is the problem that inspired me to think seriously about the advantage of an “algorithmic” construction over an explicit / naive inductive one.