35. There is a pile of 660 stones on the table. In one move, you can split any of the existing piles into two smaller ones. At the same time, the sizes of any two piles on the table at the same time must differ by less than a factor of two. What is the maximum number of piles that can result?

## 9th Grade

## Plot 1

Let $I$ be the center of the inscribed circle $\alpha$ of triangle $ABC$. The circumcircle of triangle $AIC$ intersects $\alpha$ at points $P$ and $Q$ such that $P$ and $A$ lie on the same side of line $BI$, and $Q$ and $C$ lie on the opposite side. Let $M$ be the midpoint of the smaller arc $AB$ of the circumcircle of triangle $ABC$, and $N$ be the midpoint of the smaller arc $BC$.
35. Answer: 30

Example: Let's go in reverse - we will combine two piles into one without violating the conditions of the problem. Let's say we initially had piles (two of each kind): $15,15,16,16, \ldots, 29,29$. It is easy to see that the condition is not violated, and the sum is 660. We will combine the two smallest piles (both 15), then again the two smallest (both 16), and so on until the piles of