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Choice of indexes in a sequence

Source: Kürschák 2005, problem 1

July 13, 2014
probabilityalgebra unsolvedalgebra

Problem Statement

Let N>1N>1 and let a1,a2,…,aNa_1,a_2,\dots,a_N be nonnegative reals with sum at most 500500. Prove that there exist integers k≥1k\ge 1 and 1=n0<n1<⋯<nk=N1=n_0<n_1<\dots<n_k=N such that ∑i=1kniani−1<2005.\sum_{i=1}^k n_ia_{n_{i-1}}<2005.
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