HMMT 二月 2005 · TEAM1 赛 · 第 9 题

英文解析

9. [25] Let A A . . . A be a regular k -gon inscribed in a circle of radius 1, and let P
   1 2 kbe a point lying on or inside the circumcircle. Find the maximum possible value of
   ( P A )( P A ) · · · ( P A ).
   1 2 kk − 1
   Solution: Place the vertices at the k th roots of unity, 1 , ω, . . . , ω , and place P atsome complex number p . Thenk − 1
   ∏
   2 i 2
   (( P A )( P A ) · · · ( P A )) = | p − ω |
   1 2 ki =0
   k 2 = | p − 1 | ,
   k k − 1 ksince x − 1 = ( x − 1)( x − ω ) · · · ( x − ω ). This is maximized when p is as far aspossible from 1, which occurs when p = − 1. Therefore, the maximum possible valuekof ( P A )( P A ) · · · ( P A ) is 2.
   1 2 k