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HMMT 十一月 2008 · 冲刺赛 · 第 25 题

HMMT November 2008 — Guts Round — Problem 25

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

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