Question 141
At the start of a game there are $2^m$ sheets of paper, each with a $1$ written on it. Consider the following operation: each time we choose two sheets, with numbers $a$ and $b$ on them. We erase the numbers on both sheets and write $a+b$ on both sheets. Prove that after $m2^{m-1}$ steps, the sum of the numbers on all the sheets is at least $4^m$.