HMMT 二月 2003 · GEN1 赛 · 第 9 题

英文解析

9. Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonalsintersecting only at vertices. What is the smallest possible number of obtuse trianglesin the triangulation?
   Solution: 1999
   By induction, it follows easily that any triangulation of an n -gon inscribed in a circlehas n − 2 triangles. A triangle is obtuse unless it contains the center of the circle in itsinterior (in which case it is acute) or on one of its edges (in which case it is right). Itis then clear that there are at most 2 non-obtuse triangles, and 2 is achieved when the center of the circle is on one of the diagonals of the triangulation. So the minimumnumber of obtuse triangles is 2001 − 2 = 1999. 2