HMMT 二月 2005 · TEAM1 赛 · 第 10 题

HMMT February 2005 — TEAM1 Round — Problem 10

[25] Let P be a regular k -gon inscribed in a circle of radius 1. Find the sum of the squares of the lengths of all the sides and diagonals of P . n n − 1

解析

[25] Let P be a regular k -gon inscribed in a circle of radius 1. Find the sum of the squares of the lengths of all the sides and diagonals of P . 2 k − 1 Solution: Place the vertices of P at the k th roots of unity, 1 , ω, ω , . . . , ω . We will first calculate the sum of the squares of the lengths of the sides and diagonals that contain the vertex 1. This is k − 1 k − 1 ∑ ∑ i 2 i i | 1 − ω | = (1 − ω )(1 − ω ¯ ) i =0 i =0 k − 1 ∑ i i = (2 − ω − ω ¯ ) i =0 k − 1 ∑ i = 2 k − 2 ω i =0 = 2 k, k − 1 using the fact that 1 + ω + · · · + ω = 0. Now, by symmetry, this is the sum of the squares of the lengths of the sides and diagonals emanating from any vertex. Since 2 there are k vertices and each segment has two endpoints, the total sum is 2 k · k/ 2 = k . 4 n n − 1