Task 1. (3 points) A piece of cheese is given. It is allowed to choose any positive real number $a$, different from 1, and to cut the piece in the ratio 1 : a by weight, then to cut one of the resulting pieces in the same ratio, and so on. Is it possible with a finite number of cuts to obtain pieces of cheese that can be distributed into two portions of the same weight?
Solution. Let's cut the two pieces obtained from the first cut. We will get 4 pieces whose weights are in the ratio 1:a:a:a². It is sufficient to choose $a$ such that $1+2a=a^{2}$. The positive root of this equation is

$a=1+\sqrt{2}$.