MathDB

improvement of a positive number its replacement by a power of two

Source: St. Petersburg 2019 9.5

May 2, 2019

combinatoricsSumalgebranumber theory

Problem Statement

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}