HMMT 二月 2005 · TEAM1 赛 · 第 9 题

HMMT February 2005 — TEAM1 Round — Problem 9

[25] Let A A . . . A be a regular k -gon inscribed in a circle of radius 1, and let P 1 2 k be a point lying on or inside the circumcircle. Find the maximum possible value of ( P A )( P A ) · · · ( P A ). 1 2 k

解析

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