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HMMT 十一月 2008 · 冲刺赛 · 第 25 题

HMMT November 2008 — Guts Round — Problem 25

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 13 ] Evaluate the sum
    ( ) ( ) ( )
    2 π 4 π 34 πcos + cos + · · · + cos .
    18 18 18
解析

英文解析

  1. [ 13 ] Evaluate the sum
    ( ) ( ) ( )
    2 π 4 π 34 πcos + cos + · · · + cos .
    18 18 18
    ( )
    ( )
    (18 − k ) πkπ
    Answer: − 1 If k < 18, then we can pair cos with cos , and these two terms sum to
    18 18
    ( )
    ( )
    (36 − k ) πkπ
  2. If k > 18, then the pair cos and cos also sums to 0. So, the only term in this series
    18 18
    ( )
    18 πthat is left over is cos = − 1.
    18