HMMT 二月 2004 · 冲刺赛 · 第 37 题

英文解析

37. Simplify sin(2 πk/ 4009).
    k =1
    √
    4009
    Solution:
    2004
    ∏2
    k − k
    4008
    ζ − ζ
    2 πi/ 4009 4009 k
    Let ζ = e so that sin(2 πk/ 4009) = and x − 1 = ( x − ζ ). Hencek =0
    2 i
    ∏ ∏
    4008 4008
    4008 k k
    1 + x + · · · + x = ( x − ζ ). Comparing constant coefficients gives ζ = 1,
    k =1 k =1
    ∏ ∏
    4008 4008
    k ksetting x = 1 gives (1 − ζ ) = 4009, and setting x = − 1 gives (1 + ζ ) = 1.
    k =1 k =1
    Now, note that sin(2 π (4009 − k ) / 4009) = − sin(2 πk/ 4009), so
    ( )
    2004 40082
    ∏ ∏
    2004
    sin(2 πk/ 4009) = ( − 1) sin(2 πk/ 4009)
    k =1 k =1
    4008
    k − k
    ∏
    ζ − ζ
    2 i=
    k =1
    4008
    2 k
    ∏
    1 ζ − 1
    4008 k=
    (2 i ) ζk =1
    4008
    ∏
    2 k 1 = ( ζ − 1)
    4008
    k =12
    4008
    ∏
    k k 1 = ( ζ − 1)( ζ + 1)
    4008
    k =12
    4009 · 1 = .
    4008
    However, sin( x ) is nonnegative on the interval [0 , π ], so our product is positive. Hence 122
    √
    4009
    it is .
    2004 2