MathDB

A3

Part of 2008 Balkan MO Shortlist

Problems(1)

Prove that there exists a B when there exists an A

Source: Balkan MO ShortList 2008 A3

4/6/2020
Let (am)(a_m) be a sequence satisfying an≥0a_n \geq 0, n=0,1,2,…n=0,1,2,\ldots Suppose there exists A>0A >0, am−am+1a_m - a_{m+1} ≥Aam2\geq A a_m ^2 for all m≥0m \geq 0. Prove that there exists B>0B>0 such that \begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}