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every n = sum a_ if a_k < 1 + a_1 + a2_ +... + a_{k-1

Source: 1960 Hungary - Kürschák Competition p2

October 10, 2022
number theoryIntegerSum

Problem Statement

Let a1=1,a2,a3,...a_1 = 1, a_2, a_3,...: be a sequence of positive integers such that ak<1+a1+a2+...+ak−1a_k < 1 + a_1 + a_2 +... + a_{k-1} for all k>1k > 1. Prove that every positive integer can be expressed as a sum of aia_is.
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