International Mathematics Competition
for University Students
2018

IMC2018: Day 1, 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 \(\displaystyle \dfrac{a_n}{c_n}\).

  Solution