MEMO 2011 pojedinačno problem 1
Kvaliteta:
Avg: 4,0Težina:
Avg: 6,0 Initially, only the integer
is written on a board. An integer a on the board can be re- placed with four pairwise different integers
such that the arithmetic mean
of the four new integers is equal to the number
. In a step we simultaneously replace all the integers on the board in the above way. After
steps we end up with
integers
on the board. Prove that
![44](/media/m/6/4/a/64adc94ec9a9583a35dda936791e1ef1.png)
![a_1, a_2, a_3, a_4](/media/m/a/0/7/a07dd4766d5e7c664f78e4f22e09ebfb.png)
![\frac 14 (a_1 + a_2 + a_3 + a_4)](/media/m/e/0/3/e03ef6de25bbfe66079c10794b2a349a.png)
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![30](/media/m/f/7/2/f7237ad0b4286650c3902269b3d01bdb.png)
![n = 4^{30}](/media/m/c/2/b/c2bf9d6f7e77317985db90be4f18fc24.png)
![b_1, b2,\ldots, b_n](/media/m/6/c/d/6cdc33569094a61bebf21c487967b1dc.png)
![\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.](/media/m/c/a/f/caf6b62c09fac3c6e7ef7412d6c03c8d.png)
Izvor: Srednjoeuropska matematička olimpijada 2011, pojedinačno natjecanje, problem 1