The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$. a) If $A = \mathbb{R}$, find all functions satisfying the conditions. b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions. (With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)
Problem
Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade
Tags: algebra, functional equation