IMC 1995 · Problem 2

Let $f$ be a continuous function on $[0,1]$ such that for every $x \in [0,1]$ we have $\int_x^1 f(t)\,dt \ge \dfrac{1 - x^2}{2}$. Show that $\int_0^1 f^2(t)\,dt \ge \dfrac{1}{3}$.