Question 2
Let $T$ be the sum of $k$ positive numbers $m_1,\ldots,m_k$. Prove the inequality printed in the source:
$$\sum_{k=1}^{n}\left(m_k+\frac1{m_k}\right)^2\ge k\left(\frac{k}{T}+\frac{T}{k}\right)^2.$$
Editorial transcription note: the source uses $n$ as the upper limit and reuses $k$ as the summation index, although only $k$ numbers are defined. This inconsistency is retained above; the solution explicitly treats the natural $k$-term interpretation.
$$\sum_{k=1}^{n}\left(m_k+\frac1{m_k}\right)^2\ge k\left(\frac{k}{T}+\frac{T}{k}\right)^2.$$
Editorial transcription note: the source uses $n$ as the upper limit and reuses $k$ as the summation index, although only $k$ numbers are defined. This inconsistency is retained above; the solution explicitly treats the natural $k$-term interpretation.