3

Part of 2009 Hungary-Israel Binational

Problems(2)

Does there exist a pair of strictly monotonic funct

Source: Hungary-Israel Binational Olympiad 2009, Problem 3

Does there exist a pair

(f; g)

of strictly monotonic functions, both from

\mathbb{N}

\mathbb{N}

f(g(g(n))) < g(f(n))

n \in\mathbb{N}

functioninductioninequalitiesalgebra unsolvedalgebra

Do there exist 2009 distinct positive integers

Source: Hungary-Israel Binational Olympiad 2009, Problem 6

(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers? (b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?

inductionnumber theory unsolvednumber theory