CMO 1990 Problem 3

A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions: i. for $x,y\ge 0$ , $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$ ; ii. there exists a constant $M$ ( $M>0$ ), such that $|f(x)|\le M$ when $0\le x\le 1$ . Prove that $f(x)\le x^2$ .