MathDB

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

.
Proposed by Abbas Mehrabian, Iran

C2