Problem

Source: ELMO 2015, Problem 2 (Shortlist N1)

Tags: Elmo, number theory, Combinatorial sum, Hi



Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \] Proposed by Yang Liu