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IMOC 2021 A5

Source: IMOC 2021 A5

August 12, 2021
algebra

Problem Statement

Let MM be an arbitrary positive real number greater than 11, and let a1,a2,...a_1,a_2,... be an infinite sequence of real numbers with an∈[1,M]a_n\in [1,M] for any n∈Nn\in \mathbb{N}. Show that for any ϵ≥0\epsilon\ge 0, there exists a positive integer nn such that anan+1+an+1an+2+⋯+an+t−1an+t≥t−ϵ\frac{a_n}{a_{n+1}}+\frac{a_{n+1}}{a_{n+2}}+\cdots+\frac{a_{n+t-1}}{a_{n+t}}\ge t-\epsilon holds for any positive integer tt.
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