HMMT 二月 2006 · TEAM1 赛 · 第 6 题

英文解析

6. [15] Let n be an integer at least 5. At most how many diagonals of a regular n -goncan be simultaneously drawn so that no two are parallel? Prove your answer.
   Answer: n
   Solution: Let O be the center of the n -gon. Let us consider two cases, based on theparity of n :
   • n is odd. In this case, for each diagonal d , there is exactly one vertex D of the 4
   n -gon, such that d is perpendicular to line OD ; and of course, for each vertex
   D , there is at least one diagonal d perpendicular to OD , because n ≥ 5. Theproblem of picking a bunch of d ’s so that no two are parallel is thus transmutedinto one of picking a bunch of d ’s so that none of the corresponding D ’s are the same. Well, go figure.
   • n is even. What can I say? For each diagonal d , the perpendicular dropped from
   O to d either passes through two opposite vertices of the n -gon, or else bisectstwo opposite sides. Conversely, for each line joining opposite vertices or bisectingopposite sides, there is at least one diagonal perpendicular to it, because n ≥ 6.
   By reasoning similar to the odd case, we find the answer to be n .