. A convex regular polygon with 1997 vertices has been decomposed into triangles using diagonals that do not intersect internally. How many of these triangles are acute?
. Only one of the triangles in the tiling is acute. Indeed, the fundamental remark is that acute triangles are recognized as those that have the center of their circumscribed circle as an interior point. Here, all the triangles in the tiling have the same circumscribed circle, namely the circumscribed circle of the regular polygon. It is then obvious that the center of this common circle is inside a single triangle. (It might seem a priori that the center could be located on a side that is common to two triangles, but in reality this cannot happen since the sides of the triangles are the diagonals of the regular polygon, and these do not pass through the center of the circle because the polygon has an odd number of sides.)