MathDB
Sequence Problem

Source: 2021 China Second Round Olympiad(B) P3

March 6, 2022
algebrainequalities

Problem Statement

Let {an}\{a_n\}, {bn}\{b_n\} be sequences of positive real numbers satisfying an=1100∑j=1100bn−j2a_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} b_{n-j}^2} and bn=1100∑j=1100an−j2b_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} a_{n-j}^2}
For all n≥101n\ge 101. Prove that there exists m∈Nm\in \mathbb{N} such that ∣am−bm∣<0.001|a_m-b_m|<0.001
[url=https://zhuanlan.zhihu.com/p/417529866] Link
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