MathDB

Initially, only the integer

44

is written on a board. An integer a on the board can be re- placed with four pairwise different integers

a_1, a_2, a_3, a_4

such that the arithmetic mean

\frac 14 (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, b2,\ldots, b_n

on the board. Prove that

\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.

Problem Statement