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Sequence with condition on million consecutive terms

Source: 2021 Iberoamerican Mathematical Olympiad, P3

October 20, 2021
inequalities

Problem Statement

Let a1,a2,a3,…a_1,a_2,a_3, \ldots be a sequence of positive integers and let b1,b2,b3,…b_1,b_2,b_3,\ldots be the sequence of real numbers given by b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1 Show that, if there exists at least one term among every million consecutive terms of the sequence b1,b2,b3,…b_1,b_2,b_3,\ldots that is an integer, then there exists some kk such that bk>20212021b_k > 2021^{2021}.
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