HMMT 二月 2014 · 团队赛 · 第 7 题

HMMT February 2014 — Team Round — Problem 7

[ 30 ] Find the maximum possible number of diagonals of equal length in a convex hexagon.

解析

[ 30 ] Find the maximum possible number of diagonals of equal length in a convex hexagon. Answer: 7 First, we will prove that 7 is possible. Consider the following hexagon ABCDEF √ √ √ √ 1 3 1 3 1 3 1 3 whose vertices are located at A (0 , 0), B ( , 1 − ), C ( , ), D (0 , 1), E ( − , ), F ( − , 1 − ). One 2 2 2 2 2 2 2 2 can easily verify that all diagonals but BE and CF have length 1. Now suppose that there are at least 8 diagonals in a certain convex hexagon ABCDEF whose lengths are equal. There must be a diagonal such that, with this diagonal taken out, the other 8 have equal length. There are two cases. Case I. The diagonal is one of AC, BD, CE, DF, EA, F B . WLOG, assume it is AC . We have EC = EB = F B = F C . Thus, B and C are both on the perpendicular bisector of EF . Since ABCDEF is convex, both B and C must be on the same side of line EF , but this is impossible as one of B or C , must be contained in triangle CEF . Contradiction. Case II: The diagonal is one of AD, BE, CF . WLOG, assume it is AD . Again, we have EC = EB = F B = F C . By the above reasoning, this is a contradiction. Thus, 7 is the maximum number of possible diagonals.