Question 4
Let $a_i\gt 0$ for $i=1,\ldots,n$, where $n\ge2$, and put $S=\sum_{k=1}^n a_k$. Prove that
$$\sum_{k=1}^n\frac{a_k^{2^m}}{(S-a_k)^{2^t-1}}\ge\frac{S^{1+2^m-2^t}}{(n-1)^{2^t-1}n^{2^m-2^t}}$$
for all integers $m\ge t\ge0$. When does equality hold?
$$\sum_{k=1}^n\frac{a_k^{2^m}}{(S-a_k)^{2^t-1}}\ge\frac{S^{1+2^m-2^t}}{(n-1)^{2^t-1}n^{2^m-2^t}}$$
for all integers $m\ge t\ge0$. When does equality hold?