HMMT 二月 2006 · TEAM1 赛 · 第 6 题

HMMT February 2006 — TEAM1 Round — Problem 6

[15] Let n be an integer at least 5. At most how many diagonals of a regular n -gon can be simultaneously drawn so that no two are parallel? Prove your answer.

解析

[15] Let n be an integer at least 5. At most how many diagonals of a regular n -gon can 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 the parity of n : 4 • n is odd. In this case, for each diagonal d , there is exactly one vertex D of the 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. The problem of picking a bunch of d ’s so that no two are parallel is thus transmuted into 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 bisects two opposite sides. Conversely, for each line joining opposite vertices or bisecting opposite 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 .