RU-SPB 2019 Problem 95

Call the improvement of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$ ), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$ . Prove that you can erase some of them, and improve each of the other numbers so that the sum the resulting numbers were again $2^{100}$ .