### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

Home

Results
Individuals
Teams

Official IMC site

### Problem 1

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.

(Proposed by Tomáš Bárta, Charles University, Prague)

Hint for $\displaystyle (1)\implies(2)$: Find an upper bound on $\displaystyle \displaystyle\sum_{n=1}^\infty\sqrt{\dfrac{a_n}{b_n}}$.

Hint for $\displaystyle (2)\implies(1)$: $\displaystyle \sqrt{\dfrac{a_n}{b_n}}$ is a particular case of $\displaystyle \dfrac{a_n}{c_n}$.