Bernicheyni i. A square sheet of paper was cut into two parts along a straight line. One of the resulting parts was again cut into two parts, and so on, many times. What is the minimum number of cuts needed so that among the resulting parts there could be exactly 100 twenty-sided polygons?
Answer: It is possible to obtain exactly 100 20-sided polygons with 1699 cuts, and it is impossible to obtain 100 20-sided polygons with fewer cuts. With each cut, the total number of paper pieces increases by 1 (since one piece is cut into two new pieces), so after $n$ cuts, there will be ($n+1$) pieces of paper. Now let's calculate the maximum possible total number of vertices in all pieces after $n$ cuts. With each cut, the total number of vertices increases by either 2 (if the cut goes through two vertices), 3 (if the cut goes through a vertex and a side), or 4 (if the cut goes through two sides). Since there were initially 4 vertices, after $n$ cuts, the total number of vertices in all pieces together will not exceed $4n+4$. Suppose that after $N$ cuts, 100 20-sided polygons are obtained. Since the total number of pieces after these cuts will be $N+1$, there will be $N+1-100$ additional pieces besides these 20-sided polygons. Each of these additional pieces will have at least three vertices, so the total number of vertices in all pieces will be at least $100 \cdot 20 + (N-99) \cdot 3$. As proven earlier, this number does not exceed $4N+4$. Therefore, $4N+4 \geq 100 \cdot 20 + (N-99) \cdot 3 = 3N + 1703$, from which it follows that $N \geq 1699$. Thus, we have proven that it is impossible to obtain 100 20-sided polygons with fewer than 1699 cuts. This is the main and most difficult part of the proof.

Now let's show how 100 20-sided polygons can be obtained with 1699 cuts. Here is one way: cut the square into 100 rectangles (99 cuts) and convert each rectangle into a 20-sided polygon by making 16 cuts to remove triangles from the corners (1600 cuts). In total, there will be 1699 cuts.

## Answer

It is possible to obtain exactly 100 20-sided polygons with 1699 cuts, and it is impossible to obtain 100 20-sided polygons with fewer cuts.