International Mathematics Competition
for University Students
2016

IMC2016: Day 2, Problem 6

6. Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}$. Prove that $$ \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2. $$

Proposed by Gerhard J. Woeginger, The Netherlands

Hint: Interchange the sums.

  Solution