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 $f(x)\le x^2$.
Problem
Source: China Mathematical Olympiad 1990 problem3
Tags: function, inequalities, induction, algebra unsolved, algebra