For any two positive integers $n>m$ prove the following inequality: $$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$As always, $[x,y]$ means the least common multiply of $x,y$. Proposed by A. Golovanov
Problem
Source: St. Petersburg MO 2001, 11th grade, P4
Tags: inequalities, number theory, greatest common divisor, algebra, GCD, least common multiple