2021 OIM Problems/Problem 3

Problem

Let $a_1, a_2, a_3, \cdots$ be a sequence of positive integers and let $b_1, b_2, b_3, \cdots$ be the sequence of real numbers given by

$b_n=\frac{a_1a_2,\cdots a_n}{a_1+a_2+\cdots + a_n}, \text{ for } n \ge 1.$

Show that if among every one million consecutive terms of the sequence $b_1, b_2, b_3, \cdots$ there is at least one integer, then there is some $k$ such that $b_k > 2021^{2021}$ .

Solution

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2021 OIM Problems/Problem 3

Problem

Solution

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