HMMT 二月 2014 · 团队赛 · 第 7 题

英文解析

7. [ 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 lengthsare equal. There must be a diagonal such that, with this diagonal taken out, the other 8 have equallength. 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 isconvex, 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.