Question 122
Prove that for any two positive integers $m$ and $n$ with $m \gt n$ the following inequality holds:
$$
\operatorname{lcm}(m, n) + \operatorname{lcm}(m + 1, n + 1) \gt \frac{2mn}{\sqrt{m-n}}.
$$
$$
\operatorname{lcm}(m, n) + \operatorname{lcm}(m + 1, n + 1) \gt \frac{2mn}{\sqrt{m-n}}.
$$