2014 IMO Shortlist C2

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 $m 2^{m-1}$ steps, the sum of the numbers on all the sheets is at least $4^{m}$. (Iran)

We have $2^{m}$ sheets of paper, with the number 1 written on each of them.我们执行以下操作。在每一步中，我们选择两张不同的纸； 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.证明经过$m 2^{m-1}$步后，所有纸上的数字之和至少为$4^{m}$。 (伊朗)