KMO 2017

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following.

For all nonnegative reals $x, y$ , $f(x+y^2) \ge cf(x)+y$ .

Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

For a positive integer $n$ , there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ well-formed. If the maximum number of students in a well-formed set is no more than $n$ , find the maximum number of well-formed set.

Here, an empty set and a set with one student is regarded as well-formed as well.