MathDB

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Part of 2011 Middle European Mathematical Olympiad

Problems(2)

Show that (sum b_i^2)/n is greater or equal to 2011

Source: Middle European Mathematical Olympiad 2011 - Individuals I-1

9/6/2011
Initially, only the integer 4444 is written on a board. An integer a on the board can be re- placed with four pairwise different integers a1,a2,a3,a4a_1, a_2, a_3, a_4 such that the arithmetic mean 14(a1+a2+a3+a4)\frac 14 (a_1 + a_2 + a_3 + a_4) of the four new integers is equal to the number aa. In a step we simultaneously replace all the integers on the board in the above way. After 3030 steps we end up with n=430n = 4^{30} integers b1,b2,…,bnb_1, b2,\ldots, b_n on the board. Prove that b12+b22+b32+⋯+bn2n≥2011.\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.
quadraticsnumber theory proposednumber theory
The functional equation problem of MEMO 2011 (team compt.)

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-1

9/6/2011
Find all functions f:R→Rf : \mathbb R \to \mathbb R such that the equality y2f(x)+x2f(y)+xy=xyf(x+y)+x2+y2y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2 holds for all x,y∈Rx, y \in \Bbb R, where R\Bbb R is the set of real numbers.
functionalgebra proposedalgebra