HMMT 二月 2005 · TEAM1 赛 · 第 10 题

英文解析

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 .
    2 k − 1
    Solution: Place the vertices of P at the k th roots of unity, 1 , ω, ω , . . . , ω . Wewill first calculate the sum of the squares of the lengths of the sides and diagonals thatcontain the vertex 1. This isk − 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
    ∑
    = 2 k − 2 ωii =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. Sincethere are k vertices and each segment has two endpoints, the total sum is 2 k · k/ 2 = k .2
    n n − 14