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Aime 2008 Ii 8

  1. May 20, 2008 #1

    ehrenfest

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    [SOLVED] Aime 2008 Ii 8

    1. The problem statement, all variables and given/known data
    Let [itex]a = \pi/2008[/itex]. Find the smallest positive integer n such that
    [tex]2[\cos(a)\sin(a)+\cos(4a)\sin(2a)+\cos(9a)\sin(3a)+\cdots+\cos(n^2a)\sin(na)] [/tex]
    is an integer.

    2. Relevant equations
    [tex]\cos(a+b) = \cos a \cos b- \sin a \sin b[/tex]

    [tex]\sin (a+b) = \sin a \cos b + \sin b \cos a[/tex]


    3. The attempt at a solution
    Can someone give me a hint please? This should only require high school math. I am not sure if the identities above are useful here or if there is a totally different method needed.
     
    ehrenfest, May 20, 2008
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  3. May 20, 2008 #2

    Tedjn

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    It's an AIME trig problem, which often means you have to play around with it and hope things end up canceling. You have a product which is difficult to sum, so try changing the product into a sum using the sum to product identities:

    http://www.mathwords.com/s/sum_to_product_identities.htm
     
    Tedjn, May 20, 2008
  4. May 20, 2008 #3

    ehrenfest

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    very nice
     
    ehrenfest, May 20, 2008
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