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Aime 2008 Ii 8

  • Thread starter ehrenfest
  • Start date
  • #1
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[SOLVED] Aime 2008 Ii 8

Homework Statement


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.

Homework 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]


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.
 

Answers and Replies

  • #2
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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
 
  • #3
2,012
1
very nice
 

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