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Analysis flavored inequality for monotone sequence

Source: 2021 Macedonian Team Selection Test P1

May 30, 2021
inequalities

Problem Statement

Let k≥2k\geq 2 be a natural number. Suppose that a1,a2,…a2021a_1, a_2, \dots a_{2021} is a monotone decreasing sequence of non-negative numbers such that ∑i=n2021ai≤kan\sum_{i=n}^{2021}a_i\leq ka_n for all n=1,2,…2021n=1,2,\dots 2021. Prove that a2021≤4(1−1k)2021a1a_{2021}\leq 4(1-\frac{1}{k})^{2021}a_1.
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