Middle European Mathematical Olympiad 2011 Problem I-1

Initially, only the integer $44$ is written on a board. An integer $a$ on the board can be replaced with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac{1}{4}(a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b_2, \ldots, b_n$ on the board. Prove that $$\frac{b_1^2 + b_2^2 + \ldots + b_n^2}{n} \geq 2011.$$