Difference between revisions of "2005 AIME II Problems/Problem 9"

 
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== Problem ==
 
== Problem ==
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The <math>27</math> cubes are randomly arranged to form a <math>3\times 3\times 3</math> cube. Given the probability of the entire surface area of the larger cube is orange is <math>\frac{p^a}{q^b r^c}</math> where <math>p,q,</math> and <math>r</math> are distinct primes and <math>a,b,</math> and <math>c</math> are positive integers, find <math>a+b+c+p+q+r</math>.
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For how many positive integers <math> n </math> less than or equal to 1000 is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>?
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== Solution ==
 
== Solution ==
 
== See Also ==
 
== See Also ==
 
*[[2005 AIME II Problems]]
 
*[[2005 AIME II Problems]]

Revision as of 22:26, 8 July 2006

Problem

For how many positive integers $n$ less than or equal to 1000 is $(\sin t + i \cos t)^n = \sin nt + i \cos nt$ true for all real $t$?

Solution

See Also

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