Auto: :Shapovalov A.B.

a) There is a piece of cheese. It is allowed to choose any positive (possibly non-integer) number \( a \neq 1 \), and cut this piece in the ratio \( 1: a \) by weight, then cut any of the resulting pieces in the same ratio, and so on. Is it possible to act in such a way that after a finite number of cuts, the entire cheese can be divided into two piles of equal weight?

b) The same question, but a positive rational \( a \neq 1 \) is chosen.
a) Cutting in the ratio $1: a$ is equivalent to cutting off a share of $\frac{a}{a+1}$. Choose such an $a$ that $\frac{a}{a+1}=\frac{1}{\sqrt{2}}$, and cut the largest of the pieces a second time. Then the weight of the largest of the resulting pieces is half the weight of the entire cheese.

b) Assume the opposite: after several cuts, it was possible to divide all the pieces into two equal piles. Without loss of generality, we can assume that we cut all the pieces at each step. After $k$ steps, pieces with weights in the ratio $1: a: a^{2}: \ldots: a^{k}$ were obtained, and 1 and $a^{k}$ correspond to exactly one piece each. Substituting $a$ with the irreducible fraction $m / n$ and multiplying all weights by a suitable constant, we get integer weights $n^{k}, m n^{k-1}, \ldots, m^{k}$. Since $m$ and $n$ are coprime, the weight of one pile is a multiple of $m$, while the other, which contains the piece of weight $n^{k}$, is not. Contradiction.

## Answer

a) Yes. b) No.