3

Part of 2008 Bosnia Herzegovina Team Selection Test

Problems(2)

30 persons (round table)

Source: Bosnian TST, 2008, 1.st day

30

persons are sitting at round table. 30 \minus{} N of them always speak true ("true speakers") while the other

N

of them sometimes speak true sometimes not ("lie speakers"). Question: "Who is your right neighbour - "true speaker" or "lie speaker" ?" is asked to all 30 persons and 30 answers are collected. What is maximal number

N

for which (with knowledge of these answers) we can always be sure (decide) about at least one person who is "true speaker".

combinatorics proposedcombinatorics

Nice functional equation

Source: Iran '99, Bosnian TST 2008, 2.nd day

Find all functions

f: \mathbb{R}\rightarrow \mathbb{R}

satisfying f(f(x) \plus{} y) \equal{} f(x^2 \minus{} y) \plus{} 4f(x)y for all

x,y \in \mathbb{R}