### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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Problem 1. Let $\displaystyle (a_n)_{n=1}^{\infty}$ and $\displaystyle (b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:
(1) There is a sequence $\displaystyle (c_n)_{n=1}^{\infty}$ of positive numbers such that $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \dfrac{a_n}{c_n}$ and $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \dfrac{c_n}{b_n}$ both converge;
(2) $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \sqrt{\dfrac{a_n}{b_n}}$ converges.