RO-NMO 2004 Problem 123

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that $$\int_0^1 f(x) dx = \int_0^1 x f(x) dx = 1 .$$ Prove that $$\int_0^1 f^2 (x) dx \geq 4 .$$ Ion Rasa