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Find Q for sequence with non-negative reals elements

Source: AIMO 2008, TST 4, P1, Suggested by Christian Reiher

January 4, 2009
inequalitiesalgebra unsolvedalgebra

Problem Statement

Determine Q∈R Q \in \mathbb{R} which is so big that a sequence with non-negative reals elements a1,a2,… a_1 ,a_2, \ldots which satisfies the following two conditions: (i) ∀m,n≥1 \forall m,n \geq 1 we have a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right) (ii) ∀k≥0 \forall k \geq 0 we have a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}} such that for each sequence element we have the inequality an≤Q. a_n \leq Q.
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