MathDB

2

Part of 2011 International Zhautykov Olympiad

Problems(2)

International Zhautykov Olympiad 2011 - Problem 2

Source:

1/16/2011
Find all functions f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} which satisfy the equality, f(x+f(y))=f(x−f(y))+4xf(y)f(x+f(y))=f(x-f(y))+4xf(y) for any x,y∈Rx,y\in\mathbb{R}.
functionalgebrafunctional equationalgebra unsolved
International Zhautykov Olympiad 2011 - Problem 5

Source:

1/17/2011
Let nn be integer, n>1.n>1. An element of the set M={1,2,3,…,n2−1}M=\{ 1,2,3,\ldots,n^2-1\} is called good if there exists some element bb of MM such that ab−bab-b is divisible by n2.n^2. Furthermore, an element aa is called very good if a2−aa^2-a is divisible by n2.n^2. Let gg denote the number of good elements in MM and vv denote the number of very good elements in M.M. Prove that v2+v≤g≤n2−n.v^2+v \leq g \leq n^2-n.
number theorygreatest common divisorEulerfunctionmodular arithmeticnumber theory unsolved