MEMO 2011 pojedinačno problem 1

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28. travnja 2012.
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.
Izvor: Srednjoeuropska matematička olimpijada 2011, pojedinačno natjecanje, problem 1