Question 93
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets: if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a+b$ on both sheets.
Prove that after $m2^{m-1}$ steps, the sum of the numbers on all the sheets is at least $4^m$.
Prove that after $m2^{m-1}$ steps, the sum of the numbers on all the sheets is at least $4^m$.