2014 IMO SL #C2

 Problem: 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 $4^m$ .

Solution: Consider the product of all the terms. Notice that due to the inequality$$(a+b)^2 \geq 4ab$$, the product scales up by a factor of at least four every time we perform the operation. It follows that after $m2^{m-1}$ steps, the product would be at least $4^{m2^{m-1}}$. By the AM-GM inequality, we have $$\sum_{i=1}^{2^m} a_i \geq 2^m \sqrt[2^m]{2^{m 2^m}} = (2^m)^2 = 4^m,$$ as desired. $\square$