HMMT 二月 2007 · CALC 赛 · 第 8 题
HMMT February 2007 — CALC Round — Problem 8
题目详情
英文原题
- [ 6 ] Suppose that ω is a primitive 2007 root of unity. Find 2 − 1 .
2 − ωjj =1
For this problem only, you may express your answer in the form m · n + p, where m, n, k, and p arekth npositive integers. Note that a number z is a primitive n root of unity if z = 1 and n is the smallestknumber amongst k = 1 , 2 , . . . , n such that z = 1.
解析
英文解析
- [ 6 ] Suppose that ω is a primitive 2007 root of unity. Find 2 − 1 .
2 − ωjj =1
For this problem only, you may express your answer in the form m · n + p, where m, n, k, and p arekth npositive integers. Note that a number z is a primitive n root of unity if z = 1 and n is the smallestknumber amongst k = 1 , 2 , . . . , n such that z = 1.
2006
Answer: 2005 · 2 + 1 . Note that
∑ ∏
2006
( z − ω )i
1 1
j =1 i 6 = j + · · · + =
2006 2006
z − ω z − ω ( z − ω ) · · · ( z − ω )
[ ]
2006 2005 d
2005 2004
z + z + · · · + 1
2006 z + 2005 z + · · · + 1 z − 1
d z = = ·
2006 2005 2006 2005
z + z + · · · + 1 z + z + · · · + 1 z − 1
2006 2005 2004 2007 2006
2006 z − z − z − · · · − 1 z − 1 2006 z − 2007 z + 1 = · = .
2007 2007
z − 1 z − 1 ( z − 1)( z − 1)
2006
2005 · 2 +1
Plugging in z = 2 gives ; whence the answer.
2007
2 − 1