HMMT 二月 2015 · 冲刺赛 · 第 22 题

英文解析

22. [ 14 ] Let A , A , . . . , A be distinct points on the unit circle with center O . For every two distinct
    1 2 2015
    integers i, j , let P be the midpoint of A and A . Find the smallest possible value ofij i j
    ∑
    OP .2
    1 ≤ i<j ≤ 2015 ij
    ( )
    ∑ ∑
    2015
    2015 · 2013 4056195 2 1
    Answer: OR Use vectors. | a + a | / 4 = (2 + 2 a · a ) / 4 = +
    i j i j
    4 4 2 2
    ∑ ∑ ∑
    1 2014 2015 2015 · 2013
    2 2
    ( | a | − | a | ) ≥ 2015 · − = , with equality if and only if a = 0, whichi i i
    4 4 4 4
    occurs for instance for a regular 2015-gon.
    k k
    2 2 − 2
    The identity still holds even if z − z = 0 for some k ≥ 1 used in the telescoping argument: why?
    ∏
    3 k 10
    This indeed works, since cos(2 θ ) 6 = 0: why?
    k =0
    Guts