MathDB

2

Part of 2008 Brazil National Olympiad

Problems(2)

6n colored points in a line

Source: Brazilian Math Olympiad 2008, Problem 2

10/28/2008
Let S S be a set of 6n 6n points in a line. Choose randomly 4n 4n of these points and paint them blue; the other 2n 2n points are painted green. Prove that there exists a line segment that contains exactly 3n 3n points from S S, 2n 2n of them blue and n n of them green.
combinatorial geometrycombinatorics unsolvedcombinatorics
A functional equation in positive integers

Source: Brazilian Math Olympiad 2008, Problem 5

10/28/2008
Prove that for all integers a>1 a > 1 and b>1 b > 1 there exists a function f f from the positive integers to the positive integers such that f(a\cdot f(n)) \equal{} b\cdot n for all n n positive integer.
functionalgorithmnumber theory unsolvednumber theory