sums of reciprocals Here is another tricky question: Suppose $m_1,\cdots,m_j$ and $m'_1,\cdots,m'_j$ are positive integers such that $\sum\limits_{k=1}^j m_k = \sum\limits_{k=1}^j m'_k$ and $\sum\limits_{k=1}^j m^2_k > \sum\limits_{k=1}^j m'^2_k$. What can we say about the sums of the reciprocals $\sum\limits_{k=1}^j \frac{1}{m_k}$ and $\sum\limits_{k=1}^j \frac{1}{m'_k}$. Which is larger than the other?

Try some examples.  $1 + 3 = 2 + 2$ with $1^2 + 3^2 > 2^2 + 2^2$ and $1/1 + 1/3 > 1/2 + 1/2$.  On the other hand, $2 + 5 + 12 = 9 + 9 + 1$ with $2^2 + 5^2 + 12^2 > 9^2 + 9^2 + 1^2$ and $1/2 + 1/5 + 1/12 < 1/9 + 1/9 + 1/1$. 
