HMMT 十一月 2013 · 团队赛 · 第 10 题
HMMT November 2013 — Team Round — Problem 10
题目详情
- [ 7 ] Let ω = cos + i sin . The imaginary part of the complex number 727 727 13 ( ) ∏ k − 1 k − 1 3 2 · 3 1 + ω + ω k =8 π π is equal to sin α for some angle α between − and , inclusive. Find α . 2 2
解析
- [ 7 ] Let ω = cos + i sin . The imaginary part of the complex number 727 727 13 ( ) ∏ k − 1 k − 1 3 2 · 3 1 + ω + ω k =8 π π is equal to sin α for some angle α between − and , inclusive. Find α . 2 2 13 3 12 12 π 1 − ω 1 − ω 6 6 Answer: Note that 727 = 3 − 2. Our product telescopes to = = 1 + ω , which 7 6 3 727 1 − ω 1 − ω 12 π 12 π has imaginary part sin , giving α = . 727 727