HMMT 十一月 2013 · 团队赛 · 第 10 题
HMMT November 2013 — Team Round — Problem 10
题目详情
英文原题
- [ 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 + ωk =8
π πis equal to sin α for some angle α between − and , inclusive. Find α .
2 2
解析
英文解析
- [ 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 + ωk =8
π πis equal to sin α for some angle α between − and , inclusive. Find α .
2 2
3 1213
12 π 1 − ω 1 − ω
6 6
Answer: Note that 727 = 3 − 2. Our product telescopes to = = 1 + ω , which
7 6
727 1 − ω3
1 − ω
12 π 12 πhas imaginary part sin , giving α = .
727 727