25.38*. Can a regular triangle be cut into 1000000 convex polygons such that any straight line intersects no more than 40 of them?
25.38. If cuts are made close to the vertices of a convex $n$-gon, then one can cut off $n$ triangles from it and obtain a convex $2n$-gon. It is easy to verify that in this case any line intersects no more than two of the cut-off triangles.

Cut off 3 triangles from an equilateral triangle, then 6 triangles from the resulting hexagon, and so on, until a $3 \cdot 2^{19}$-gon is obtained. Any line can intersect no more than two triangles cut off at each step. Therefore, a line can intersect no more than $1 + 2 \cdot 19 = 39$ polygons in total. The total number of polygons into which the equilateral triangle is divided is $1 + 3 + 3 \cdot 2 + \ldots + 3 \cdot 2^{18} = 1 + 3(2^{19} - 1) > 2^{20} = (2^{10})^2 > 1000^2$. It is clear that one can cut off not all triangles to obtain exactly 1,000,000 polygons.